Erich Friedman, PhD, of Stetson University and His Friedman Numbers

Math Magic A Friedman number is a positive integer which can be written in some non-trivial way using its own digits, together with the symbols + - x / ^ ( ) and concatenation. For example, 25 = 52 and 126 = 21 * 6. The Friedman numbers are sequence A036057 of the Encyclopedia of Integer Sequences.

All Friedman numbers with 4 or fewer digits are known. They were determined by a computer search. In fact they are:

 25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024,
1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349,
2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592 ,2737, 2916, 3125,
3159, 3281, 3375, 3378, 3685, 3784, 3864, 3972, 4088, 4096, 4106, 4167, 4536, 4624,
4628, 5120, 5776, 5832, 6144, 6145, 6455, 6880, 7928, 8092, 8192, 9025, 9216, 9261.
Determine why each of these is a Friedman number. You might also have fun confirming that 123456789 and 987654321 are Friedman numbers. Find some more Friedman numbers. (I'm especially interested in knowing all the 5-digit ones.)

What else can you show about Friedman numbers? If F(n) is the number of Friedman numbers less than n, can you show F(n)/n --> 1? or even disprove F(n)/n --> 0?

ANSWERS

Mike Reid, Ulrich Schimke, and Philippe Fondanaiche solved all the 4-digit Friedman numbers. Here are the solutions:

25 = 52 121 = 112 125 = 51+2 126 = 6 * 21 127 = - 1 + 27 128 = 28-1
153 = 3 * 51 216 = 62+1 289 = (8 + 9)2 343 = (3 + 4)3 347 = 73 + 4 625 = 56-2
688 = 8 * 86 736 = 7 + 36 1022 = 210 - 2 1024 = (4 - 2)10 1206 = 6 * 201 1255 = 5 * 251
1260 = 6 * 210 1285 = (1 + 28) * 5 1296 = 6(9-1)/2 1395 = 15 * 93 1435 = 35 * 41 1503 = 3 * 501
1530 = 3 * 510 1792 = 7 * 29-1 1827 = 21 * 87 2048 = 84 / 2 + 0 2187 = (2 + 18)7 2349 = 29 * 34
2500 = 502 + 0 2501 = 502 + 1 2502 = 2 + 502 2503 = 502 + 3 2504 = 502 + 4 2505 = 502 + 5
2506 = 502 + 6 2507 = 502 + 7 2508 = 502 + 8 2509 = 502 + 9 2592 = 25 * 92 2737 = (2 * 7)3 - 7
2916 = (1 * 6 * 9)2 3125 = (3 + 1 * 2)5 3159 = 9 * 351 3281 = (38 + 1) / 2 3375 = (3 + 5 + 7)3 3378 = (7 + 8)3 + 3
3685 = (36 + 8) * 5 3784 = 8 * 473 3864 = 3 * (- 8 + 64) 3972 = 3 + (9 * 7)2 4088 = 84 - 8 - 0 4096 = (4 + 0 * 9)6
4106 = 46 + 10 4167 = 46 + 71 4536 = 56 * 34 4624 = (64 + 4)2 4628 = 682 + 4 5120 = 5 * 210
5776 = 767-5 5832 = (2 * 5 + 8)3 6144 = 6 * 44+1 6145 = 6 * 45 + 1 6455 = (64 - 5) * 5 6880 = 8 * 860
7928 = 892 - 7 8092 = 902 - 8 8192 = 8 * 29+1 9025 = 952 + 0 9216 = 1 * 962 9261 = 219-6

Mike Reid and Philippe Fondanaiche found that 123456789 = ((86 + 2 * 7)5 - 91) / 34 and 987654321 = (8 * (97 + 6/2)5 + 1) / 34.

Mike Reid thought that Friedman numbers are "nicer" if the digits are used in the proper order. These are shown in red above and below.

Recently, I finally wrote a program capable of finding all the 5-digit Friedman numbers. Here they are:

10192 = 1012 - 9 10201 = 1012 + 0 10251 = 51 * 201 10255 = 5 * 2051 10368 = 8 * 61+0+3
10426 = 26 * 401 10521 = 21 * 501 10525 = 5 * 2105 10575 = 15 * 705 10824 = 1042 + 8 (TG)
10935 = 15 * 93 + 0 11025 = (110 - 5)2 11163 = 3 * 611+1 11259 = 9 * 1251 11264 = 11 * 26+4 (MR)
11439 = 9 * 31 * 41 11663 = 16 * 36 - 1 11664 = 1 * 1 * 66 / 4 11665 = 66 / (5 - 1) + 1 11844 = 84 * 141
11848 = 8 * 1481 11943 = 9 * (113 - 4) 12006 = 6 * 2001 12060 = 6 * 2010 12091 = 1102 - 9 (PF)
12100 = 1102 + 0 12101 = 1102 + 1 12102 = 1102 + 2 12103 = 1102 + 3 12104 = 1102 + 4
12105 = 1102 + 5 12106 = 1102 + 6 12107 = 1102 + 7 12108 = 1102 + 8 12109 = 1102 + 9
12167 = (16 + 7)1+2 (PF) 12288 = (1+2) * 88/2 12321 = (113 - 2)2 (TG) 12337 = 73 * 132 12384 = 3 * 4128
12493 = (4 + 9) * 312 12505 = 5 * 2501 12544 = (51 - 2) * 44 12546 = 51 * 246 12550 = 5 * 2510
12595 = 5 * 2519 12600 = 6 * 2100 12762 = 6 * 2127 12768 = 8 * 21 * 76 12769 = (96 + 17)2
12798 = 2 * 79 * 81 12802 = 2 * (802 + 1) 12843 = 3 * 4281 12850 = (-1 + 28) * 50 12955 = 5 * 2591
12960 = 160 * 92 12964 = 14 * 926 12996 = (6 * (1 + 9 + 9))2 13125 = 21 * 53+1 13176 = 61 * (7 - 1 )3
13225 = (1 * 5 * 23)2 13286 = 26 * (83 - 1) 13243 = 41 * 323 13496 = 4 * ((6 + 9)3 - 1) 13545 = 3 * 4515
13689 = (9 * 13)8-6 13725 = 5 * ((2 * 7)3 + 1) 13764 = 6 * 31 * 74 13813 = (3 * 8)3 - 11 13822 = (3 * 8)2+1 - 2
13823 = (3 * 8)3 - 2 + 1 13824 = (3 * 8)4+1-2 13825 = 1 + (3 * 8)-2+5 13826 = (18 + 6)3 + 2 13832 = (3 + 21)3 + 8
13842 = 243 + 18 13950 = 15 * 930 14035 = 35 * 401 14129 = 114 - 29 14168 = 11 * (64 - 8)
14175 = 7 * 451+1 14256 = (2 * 5 + 1) * 64 14350 = 35 * 410 14352 = 23 * (54 - 1) 14641 = (1+4+6)4*1 (PF)
14645 = (5 + 6)4 + 4*1 14647 = (4 + 7)4 + 6*1 15003 = 3 * 5001 15030 = 3 * 5010 15125 = 5 * (5 * 11)2
15246 = 6 * 2541 15300 = 3 * 5100 15345 = 3 * 5 * (45 - 1) 15378 = 7 * (5 + 8)3 - 1 15379 = 7 * (5 + 9 - 1)3
15435 = 3 * 5145 15495 = 15 * (45 + 9) 15552 = (15 + 5)5 * 2 15562 = 1 * 2 * (65 + 5) 15567 = 56 - 57 - 1
15568 = 56 - 58 + 1 15585 = 1 * (55 - 8) * 5 15586 = 56 - 5 * 8 + 1 15612 = - 1 + 56 - 12 15613 = 1 + 56 - 13
15615 = 56 - 5 * (1 + 1) 15617 = -1 + 56 - 1 * 7 15618 = 1 + 56 - 1 * 8 15620 = 56 - 10 / 2 (TG) 15621 = -1 + 56 - 2 - 1
15622 = 1 + 56 - 2 - 2 15623 = -1 + 56 + 2 - 3 15624 = 1 + 56 + 2 - 4 15625 = 56 * 125 15626 = 1 + 562/6
15627 = 56 + 2 * 17 15628 = 56 + 8 / 2 - 1 15629 = 56 + (9 - 1) / 2 15631 = 56 + (1 + 1) * 3 15632 = 1 + 56 + 3 * 2
15633 = - 1 + 56 + 3 * 3 15634 = 56 + 13 - 4 15635 = 56 + 5 * (3 - 1) 15641 = 56 + 41+1 15642 = 1 + 56 + 42
15645 = 1* 56 + 4 * 5 15655 = 1 * 5 * (6 + 55) 15656 = 1 + 56 + 5 * 6 15661 = 56 + 61+1 15662 = 1 + 56 + 62
15667 = 1 * 56 + 6 * 7 15679 = 56 + 9 * (7 - 1) 15688 = -1 + 56 + 8 * 8 15689 = 56 + 8 * (9 - 1) 15697 = 56 + 9 * (7 + 1)
15698 = 1 + 56 + 9 * 8 15795 = 9 * 1755 15975 = 5 * 5 * 9 * 71 16225 = 6 * 522 + 1 16245 = 5 * (61 - 4)2
16272 = 6 * 2712 16295 = (1 + 6)5 - 29 16347 = 47 - 36 - 1 16348 = 48-1 - 36 16368 = 8 * 31 * 66
16372 = (1 + 3)7 - 6 * 2 16374 = 47 - 1 - 6 - 3 16375 = (5 - 1)7 - 6 - 3 16377 = (1 + 6 - 3)7 - 7 16378 = (8 - 3 - 1)7 - 6
16381 = (1 + 1)6+8 - 3 16382 = (3 - 1)6+8 - 2 16384 = 163 * (8 - 4) (TG) 16385 = (5 - 3)6+8 + 1 16447 = - 1 + 64 + 47
16448 = 48-1 + 64 16479 = 47 + 96 - 1 16743 = 76-1 - 43 16758 = 75 - 6 * 8 - 1 16759 = 75 - 6 * (9 - 1)
16765 = 75 - 6 * (6+1) 16783 = 76-1 - 8 * 3 16794 = 76-1 - 9 - 4 16797 = 76 / 7 - 9 - 1 16798 = 76 / (8 - 1) - 9
16807 = 76-1 + 0 * 8 16815 = (1 + 6)5 + 1 * 8 16875 = 1 * 68 + 75 16879 = 76-1 + 8 * 9 17253 = (72 - 1) * 35
17325 = 75 * 231 17328 = 8 * (37 - 21) 17346 = 6 * 7 * 413 17368 = 8 * (37 - 16) 17384 = 8 * (37 - 14)
17428 = 2 * 8714 17437 = 47 * 371 17482 = 2 * 8741 17488 = 8 * (4 - 1)7 - 8 17536 = 1 + 75 + 36
17689 = (7 * 19)8-6 17856 = 8 * (56 - 1) / 7 17892 = 9 * 28 * 71 17920 = 70 * 29-1 17925 = 5 * (7 * 29 + 1)
18225 = 81 * 225 18265 = 65 * 281 18270 = 21 * 870 18432 = 18 * 43+2 (MR) 18435 = 18 * 45 + 3
18522 = 2 * 218-5 18594 = 18 * (45 + 9) 18723 = 3 * (71 + 8)2 18744 = 71 * (44 + 8) 19026 = 21 * 906
19215 = 21 * 915 19321 = 1 * 1392 19392 = 39 - 291 19453 = 19 * 45 - 3 (MR) 19592 = (5 - 2)9 - 91
19629 = (1 + 2)9 - 6 * 9 19642 = (6 / 2)9 - 41 19653 = 39 - 1 * 5 * 6 19682 = (6 / 2)9 - 18 19683 = 1 * (9 - 6)8 * 3
19684 = (6 * 4 / 8)9 + 1 19692 = (6 / 2)9 + 9 * 1 19693 = (6 - 3)9 + 1 + 9 19732 = 39 + 1 * 72 19734 = 3 * (94 + 17)
19736 = 9 * (37 + 6) - 1 19737 = 9 * (37 + 7 - 1) 19738 = 39 + 7 * 8 - 1 19739 = (- 1 + 9) * 7 + 39 19773 = 9 * (7 + 7 - 1)3
19845 = 5 * 49 * 81 20485 = 5 * (20 + 84) 20736 = (2 * 6)7-3 + 0 21175 = 7 * (5 * 11)2 21375 = 3 * 7125
21495 = 21 * 45 - 9 21504 = 21 * 45 + 0 21586 = 86 * 251 21606 = 6 * (602 + 1) 21753 = 3 * 7251
21843 = (48 - 1) / 3 - 2 21844 = (48 - 4) / (1 + 2) 21845 = (48 - 1) / (5 - 2) 21848 = (48 + 8) / (1 + 2) 21870 = 27 * 810
21875 = 7 * (8 - 2 - 1)5 21943 = (2 * 14)3 - 9 21952 = (29 - 1)5-2 (TG) 21953 = (2 * (5 + 9))3 + 1 22264 = 46 * 222
22528 = 22 * (8 / 2)5 22757 = 7 * (572 + 2) 23326 = 3 * 62+3 - 2 23328 = (2 * 33)2 * 8 23392 = 32 * (93 + 2)
23456 = 25 * (36 + 4) 23490 = 290 * 34 23546 = 23 * 45 - 6 23548 = (3 + 4) * 582 23552 = 23 * 25+5
23796 = 6 * (632 - 3) 24336 = (4 * (36 + 3))2 24339 = (4 * 39)2 + 3 24367 = 7 * (63 - 4)2 24375 = (37 + 2) * 54
24385 = (58 / 2)3 - 4 24389 = (2 * 8 + 9 + 4)3 24390 = 293 + 40 24393 = (3 * 9 + 2)3 + 4 24546 = (2 + 4) * (-5 + 46)
24564 = 6 * (4 * 45 - 2) 24566 = 6 * 46 - 2 * 5 24576 = (2 / 4)-5-7 * 6 24584 = 24 * 45 + 8 24586 = 6 * 84 + 2 * 5
24768 = 4 * 72 * 86 24964 = (94 + 64)2 24972 = 4 * (792 + 2) 25105 = 5 * 5021 25137 = 513 * 72
25314 = (154 + 3) / 2 25375 = 35 * 725 25474 = 47 * 542 25510 = 5 * 5102 25725 = 525 * 72
25872 = 528 * 72 25895 = 5 * ((8 * 9)2 - 5) 25921 = (159 + 2)2 26238 = 2 * 2 * 38 - 6 26244 = (2 / 6)-2*4 * 4 (MR)
26348 = 4 * (38 + 26) 26364 = 263 * 6 / 4 26496 = 9 * 46 * 26 26624 = 26 * 24+6 26754 = 546 * 72
26896 = (96 + 68)2 26973 = 37 * 96/2 27436 = (6 * 7 - 4)3 / 2 27634 = 2 * ((6 * 4)3 - 7) 27639 = 27 * 63 - 9
27648 = (7 - 2 / 8) * 46 27653 = 63 * 27 + 5 27654 = 27 * 45 + 6 27783 = (3 * 7)8/2 / 7 27889 = (79 + 88)2
28217 = (21 * 7)2 - 7 28224 = (2 + 82)2 * 4 28226 = (28 * 6)2 + 2 28322 = 2382 / 2 28476 = 7 * (46 - 28)
28547 = (8 + 5)4 - 7 * 2 28554 = (8 + 5)4 - 5 - 2 (TG) 28556 = (8 + 5)6-2 - 5 28559 = -2 + (8 + 5)-5+9 28561 = 1 * (8 + 5)6-2 (TG)
28564 = (8 + 5)4 + 6 / 2 (TG) 28671 = (2 / 8)-6 * 7 - 1 28672 = 7 * (8 - 2 - 2)6 28674 = 7 * (8 - 4)6 + 2 28678 = 7 * (8 * 8)2 + 6
28728 = 7 * (82*2 + 8) 28749 = 7 * (84 + 9 + 2) 28764 = 6 * (2 * 74 - 8) 28784 = 7 * (84 + 16) 28900 = (80 + 90)2 (TG)
29160 = 10 * (6 * 9)2 29184 = 4 * 8 * 912 29282 = 2 * (9+2)8/2 (MR) 29517 = (95 - 1) / 2 - 7 29519 = (95 - 9) / 2 - 1
29523 = 95 / 2 - 3 / 2 29524 = (2 * 95 - 2) / 4 29525 = (95 - 5) / 2 - 2 29526 = (95 + 6 / 2) / 2 29527 = (95 + 7 - 2) / 2
29529 = 95 / 2 + 9 / 2 29531 = (95 + 13) / 2 29549 = (95 + 49) / 2 29584 = (4 * 5 * 9 - 8)2 29632 = 32 * 926
29768 = 8 * (9 * 6 + 7)2 29795 = 59 * (29 - 7) 29929 = (9 * 9 + 92)2 30625 = (3 * 60 - 5)2 31250 = 10 * (2 + 3)5
31252 = 2 * (52*3 + 1) 31256 = 1 * 2 * (56 + 3) 31346 = 43 * 36 - 1 31347 = 43 * 37-1 31509 = 9 * 3501
31590 = 9 * 3510 31682 = 62 * (83 - 1) 32685 = (6 + 2)5 - 83 32697 = 63 * (29 + 7) 32744 = 2 * (47 - 3 * 4)
32747 = 2 * 47 - 21 32751 = 2 3*5 - 17 32759 = (3 - 2 + 7)5 - 9 32761 = 23(6-1) - 7 (TG) 32762 = 23(7-2) - 6 (TG)
32764 = 23*7-6 - 4 32765 = -3 + (2 * 7 - 6)5 (TG) 32768 = (3 - 2 + 7)6 / 8 32771 = 3 + 27+7+1 (MR) 32772 = 2 * ((7 - 3)7 + 2)
32775 = (7 + 3 - 2)5 + 7 32778 = 27+8 + 7 + 3 32781 = 27+8 + 13 32782 = 83+2 + 7 * 2 32783 = 323 + 7 + 8
32785 = 3 + 2 + 7 * 85 32786 = 27+8 + 3 * 6 32795 = 5 * (97-3 - 2) 32805 = 5 * (38 + 2 * 0) 32815 = 5 * (38 + 2 * 1)
32825 = 5 * (38 + 2 * 2) 32832 = 323 + 82 32835 = 5 * (38 + 2 * 3) 32836 = 323 + 68 32845 = 5 * (38 + 2 * 4)
32849 = 323 + 92 32851 = 215 + 83 32853 = 323 + 85 32854 = 85 + 43 * 2 32855 = 5 * (38 + 2 * 5)
32859 = 85 + 93 - 2 32865 = 5 * (38 + 2 * 6) 32875 = 5 * (38 + 2 * 7) 32885 = 5 * (38 + 2 * 8) 32895 = 5 * (38 + 2 * 9)
33495 = 33 * (45 - 9) 33579 = 7 * 9 * 533 33655 = 53 * 635 33696 = 36 * 936 34425 = 34 * 425
34968 = 3 * (9 * 64 - 8) 34986 = 48 * 93 - 6 34991 = (9 + 9)4 / 3 - 1 34992 = 3 * (9 * 2)4 / 9 34993 = ((9 + 9)4 + 3) / 3
34996 = 6 * (9 + 9)3 + 4 35152 = 2 * (5 * 5 + 1)3 35684 = 85 - 4 * 36 35721 = 35 * 7 * 21 35726 = 72 * 36 + 5
35782 = (57 - 38) / 2 35928 = (52 + 8)3 - 9 35932 = (3 * (9 + 2))3 - 5 35933 = 333 + 5 - 9 (TG) 35937 = (35 + 7 - 9)3
35942 = (42 - 9)3 + 5 (TG) 36457 = (7 * 56 - 4) / 3 36549 = 9 * (46 - 35) 36850 = (36 + 8) * 50 36855 = 63 * 585
36864 = (6 + 6 - 3) * 84 36918 = 9 * (83+1 + 6) 37179 = 37 * (1 + 7 + 9) 37187 = 17 * 37 + 8 37249 = (3 * 7 * 9 + 4)2
37449 = (49 - 4 + 3) / 7 37668 = 6 * 73 * 86 37814 = 74 * (83 - 1) 37840 = 8 * 4730 37845 = 87 * 435
37875 = 75 * (83 - 5) 38416 = 148*3/6 (TG) 38424 = (2 * (3 + 4))4 + 8 38427 = (2 * 7)4 + 8 + 3 38637 = (8 * 7 - 3) * 36
38640 = 30 * (-8 + 64) 38856 = (38 - 85) * 6 38912 = 38 * 29+1 39216 = ((9 - 2)6 - 1) / 3 39283 = 39 * 2 - 83
39288 = 8 * ((9 + 8)3 - 2) 39294 = 2 * (39 - 36) 39295 = (52 + 9)3 - 9 39304 = 343 + 0 * 9 (TG) 39313 = (33 + 1)3 + 9 (TG)
39314 = 343 + 1 + 9 39328 = 2 * 39 - 38 39342 = (39 - 3 * 4) * 2 39343 = 39 + 343 (TG) 39356 = 6 * (39 - 5) / 3
39358 = 39 * (-3 + 5) - 8 39362 = 6 * (39 - 2) / 2 39363 = 39 / 3 * 6 - 3 (MR) 39366 = 39 / (-3 + 6) * 6 (MR) 39368 = 6 * (38 + 1 / 3)
39369 = 3 + 93 * 6 * 9 39372 = (3 + 9 * 37) * 2 39382 = ((3 * 9)3 + 8) * 2 39424 = 29 * (34 - 4) 39456 = 6 * (94 + 3 * 5)
39784 = 8 * 4973 39864 = 6 * (94 + 83) 39945 = 39 * 45 + 9 41323 = 43 * 312 41468 = 4 * (8 * 64 - 1)
41472 = 2 * (1 + 4 + 7)4 42025 = 2054-2 (TG) 42336 = 6 * (34 + 3)2 42875 = (42-7)8-5 (TG) 42898 = 89 * 482
43264 = (63 - 4 - 4)2 43268 = (63 - 8)2 + 4 43375 = 53 * (73 + 4) 43688 = 86 * (83 - 4) 43689 = (49 + 8) / 6 - 3
43691 = 49 / 6 + 1 / 3 43692 = (49 + 23) / 6 43775 = (4 * 37 + 7) * 5 43932 = 3 * ((9 + 2)3 + 3) 44375 = 54 * (43 + 7)
44676 = 6 * 7446 44977 = (7 + 7)4 + 94 45056 = (50 - 6) * 45 45360 = 35 * 64 + 0 45361 = 35 * 64 + 1
45362 = 35 * 64 + 2 45363 = 35 * 64 + 3 45364 = 35 * 64 + 4 45365 = 35 * 64 + 5 45366 = 35 * 64 + 6
45367 = 35 * 64 + 7 45368 = 35 * 64 + 8 45369 = 35 * 64 + 9 45632 = -45 + 63*2 45684 = 54 * 846
45760 = 65 * 704 45864 = 84 * 546 45873 = 7 * 38 - 54 45927 = ((4 + 5) * 9)2 * 7 45947 = 4 * 5 + 94 * 7
45957 = 7 * (94 + 5) - 5 45978 = 7 * (94 + 8) - 5 46256 = 66 - (4 * 5)2 46368 = 36 * (64 - 8) 46556 = 66 - 4 * 5 * 5 (TG)
46593 = 3 * (56 - 94) 46608 = 66 - 48 + 0 (TG) 46613 = 66 - 43 * 1 (TG) 46615 = 6 * 65 - 41 46619 = 66 - 9 * 4 - 1 (TG)
46624 = 66 - 4 * 4 * 2 (TG) 46626 = -4 + 66 - 26 46630 = 4 + 66 - 30 (MR) 46632 = -4 * 6 + 63*2 46633 = 4 + 66 - 33 (TG)
46635 = 6 * (65 - 4) + 3 46637 = 66 - 4 * 3 - 7 (TG) 46640 = 66 - 4 * 4 + 0 (TG) 46641 = 66 - 4 * 4 + 1 (TG) 46642 = 66 - 4 * 4 + 2 (TG)
46643 = 66 - 4 * 4 + 3 (TG) 46644 = 4 + 66 - 4 * 4 46645 = 66 - 4 * 4 + 5 (TG) 46646 = 66 - 4 * 4 + 6 (TG) 46647 = 66 - 4 * 4 + 7 (TG)
46648 = 4 * 66 / 4 - 8 46649 = 66 - 4 * 4 + 9 (TG) 46650 = 66 - 5 - 40 46651 = -4 + 6 * 65 - 1 (TG) 46652 = -4 + (6 * 6)5-2
46653 = 66 - (5 + 4)/3 (TG) 46655 = 4 + 6 * 65 - 5 (TG) 46656 = (-4 * 6 + 6 * 5)6 46657 = 67 / 6 + 5 - 4 (TG) 46658 = 6 * 65 + 8 / 4 (TG)
46660 = 4 + 66 + 6 * 0 (TG) 46661 = 66 + 4 + 16 46662 = 62 * 64 + 6 46663 = 4 + 66 + 6 - 3 (TG) 46664 = 66 + 4 * (6 - 4) (TG)
46665 = 6 * (65 + 6 / 4) 46668 = 66 + 6 * 8 / 4 (TG) 46672 = 67 / 6 + 42 (TG) 46673 = -4 + 66 + 7 * 3 (TG) 46677 = 66 + 4 * 7 - 7 (TG)
46684 = -4 + 66 + 8*4 (TG) 46688 = (4 + 66 / 8) * 8 46691 = 66 + 4 * 9 - 1 (TG) 46851 = (4 - 1) * (56 - 8) 46875 = (4 + 7 - 8) * 56
47538 = 57 * 834 47652 = 76 * (54 + 2) 48672 = 78 * 624 48750 = 78 * 54 + 0 48751 = 78 * 54 + 1
48752 = 78 * 54 + 2 48753 = 78 * 54 + 3 48754 = 78 * 54 + 4 48755 = 78 * 54 + 5 48756 = 78 * 54 + 6
48757 = 78 * 54 + 7 48758 = 78 * 54 + 8 48759 = 78 * 54 + 9 49152 = (4 - 1) * 29+5 49277 = 9 * 742 - 7
49584 = 48 * (45 + 9) 49855 = 59 * 845 49896 = 6 * 84 * 99 49968 = 8 * 9 * 694 51200 = 50 * 210
51398 = (59 - 1) / 38 51759 = 9 * 5751 52168 = 8 * 6521 52429 = (49 + 2 / 2) / 5 52483 = 2 * 4 * 38 - 5
52488 = (5 + 2 - 4)8 * 8 52493 = 23 * 94 + 5 52498 = 8 * 94 + 2 * 5 52731 = 217 * 35 52947 = 49 / 2 - 57
53245 = 52 * 45 - 3 53248 = 52 * 48-3 53297 = 2 * 75 + 39 53824 = (8 * (34 - 5))2 53865 = 63 * 855
54369 = (3 + 4) * (65 - 9) 54378 = 87 * 54 + 3 54432 = (4 + 3) * (4 + 2)5 54436 = (4 + 3) * 65 + 4 54476 = 7 * 65 + 44
54642 = 42 * (64 + 5) 54726 = 7 * (65 + 42) 54768 = 7 * (65 + 48) 54872 = (8 * 5 - 2)7-4 54953 = 95 - (3 + 5)4
54958 = 95 - 84 + 5 55225 = (5 * (52 - 5))2 55296 = 54 * 210 56295 = 9 * 6255 56628 = (5 + 8) * 662
56732 = 26 * (37 - 5) 56875 = 65 * 875 57288 = 8 * (25+8 - 7) 57644 = 4 * (6 * 74 + 5) 57645 = 57 - 5 * 46
58921 = 95 - 28-1 58957 = 95 - 28-1 58971 = 95 - 78 * 1 58973 = 95 - 83 + 7 58978 = 95 - 8 * 8 - 7
59032 = 95 - 20 + 3 59038 = 95 - 8 - 3 - 0 59039 = 95 - 87 - 5 59044 = 95 - 4 - 40 59045 = 95 - 4 - 0 * 5
59046 = 95 - 4 + 06 59048 = 95 - 480 59049 = 95 + 0 * 4 * 9 59050 = 95 + 50 + 0 59051 = 95 + 50 + 1
59052 = 5 + 90+5 - 2 59053 = 95 + 50 + 3 59054 = 95 + 50 + 4 59055 = 95 + 50 + 5 59056 = 95 + 50 + 6
59057 = 95 + 50 + 7 59058 = 95 + 50 + 8 59059 = 95 + 50 + 9 59064 = 95 + 60 / 4 59094 = 9 * (94 + 1) + 0
59128 = 95 + 81 - 2 59129 = 95 + 92 - 1 59147 = 95 + 7 * 14 59263 = 95 + 63 - 2 59265 = 95 + 65-2
59273 = 95 + 7 * 32 59313 = 393 - 5 - 1 (TG) 59314 = 394-1 - 5 (TG) 59318 = 398-5 - 1 (TG) 59319 = 399-5-1 (TG)
59375 = (9 + 3 + 7) * 55 59392 = 95 + (9 - 2)3 59409 = 95 + 4 * 90 59451 = 19 * (55 + 4) 59759 = ((5 + 9)5 + 7) / 9
61435 = 5 * (3 * 64 - 1) 61440 = 60 * 44+1 62476 = 6 * ((7 - 2)6 - 4) 62503 = (503 + 6) / 2 62504 = 4 * (56 + 20)
62564 = 4 * 56 + 26 62968 = 68 * 926 63478 = 48 - 6 73 63895 = 65 * 983 63904 = 403 - 96
63945 = 63 * (-9 + 45) 63985 = (8 * 5)3 - 6 - 9 63994 = (49 - 9)3 - 6 64036 = 403 + 6 * 6 (TG) 64512 = 45 * (26 - 1)
64513 = 63 * 45 + 1 64522 = 2542 + 6 (TG) 64550 = (64 - 5) * 50 64868 = 48 - 668 65344 = 64 * (45 - 3)
65471 = -65 + 47+1 65478 = 48 - 65 + 7 65480 = 48 - 56 + 0 65481 = 48 - 56 + 1 65482 = 48 - 56 + 2
65483 = 48 - 56 + 3 65484 = 48 - 56 + 4 65485 = 48 - 56 + 5 65486 = 48 - 56 + 6 65487 = 48 - 56 + 7
65488 = 48 - 56 + 8 65489 = 48 - 56 + 9 65491 = 169-5 - 45 65528 = 25+5+6 - 8 (TG) 65531 = (5 - 3)16 - 5 (TG)
65536 = (6 / 3)6+5+5 (TG) 65542 = 45+5-2 + 6 65841 = 48 + 5 * 61 65884 = 48 + 6 * 58 66339 = (6 * 6)3 + 39
66554 = 65 * 45 - 6 67149 = 9 * 7461 67228 = 28 * 76-2 67234 = 6 + 72+3 * 4 67252 = 2 * 2 * (75 + 6)
67254 = 4 * (75 + 6) + 2 67392 = 72 * 936 67950 = 75 * 906 68644 = (44 + 6)8-6 68800 = 8 * 8600
69253 = 95 * 36 - 2 69255 = 95 * (5 - 2)6 69472 = 67 / 4 - 29 69822 = 862 * 92 69895 = 9 * (65 - 9) - 8
69975 = 67 / (9 - 5) - 9 69984 = 6-9/9+8 / 4 69985 = 9 * 65 + 9 - 8 69993 = 96 * 93 + 9 70225 = (270 - 5)2
71199 = 9 * 7911 72576 = 567 * 27 73125 = 13 * 752 73926 = 6 * 9 * 372 73984 = (8 * 34)9-7
74183 = 31 * (74 - 8) 74353 = 34 * 37 - 5 74358 = 34 * (8 - 5)7 74533 = 73 * (45 - 3) 74536 = 56 * (4 + 7)3
74892 = (4 + 8) * 792 74897 = (87 / 4 + 9) / 7 75433 = 47 + (3 * 3)5 76335 = 35 * (37 - 6) 76832 = 2 * (6 + 8)7-3
76835 = (6 + 8)5 / 7 + 3 77459 = 57 - 9 * 74 78055 = 58 / 5 - 70 78115 = 57 - 8 - 1 - 1 78116 = (6 - 1)7 - 8 - 1
78117 = (6 - 1)7 - 8 * 1 78123 = (8 - 3)7 - 2 * 1 78125 = 57 * 182 78126 = (8 - 6 / 2)7 + 1 78132 = (2 + 3)7 + 8 - 1
78133 = (3 + 3 - 1)7 + 8 78135 = 57 + 8 + 3 - 1 78136 = (6 - 1)7 + 8 + 3 78152 = 57 + 28 - 1 78163 = (6 - 1)7 + 38
78165 = 57 + 8 * (6 - 1) 78225 = 57 + (2 * 5)2 78545 = 57 + 5 * 84 78605 = 57 + 6 * 80 78659 = 57 + 6 * 89
78732 = (7 + 7 - 2) * 38 78975 = 9 * 8775 79299 = 92 * 979 81225 = 1 * 2852 (TG) 81648 = (8 * 8 - 1) * 64
81920 = 80 * 29+1 82372 = 2872 + 3 82755 = 5 * (75 - 28) 82936 = (3 * 96)2 - 8 82942 = (4 * 8 * 9)2 - 2
82944 = (9 * 44 / 8)2 82952 = (9 * 25)2 + 8 83357 = 73 * 25 + 8 83521 = (25 - 8)3+1 (TG) 83524 = (25 - 8)4 + 3 (TG)
83957 = 57 + 8 * 93 84375 = 5 * (7 + 8)4 / 3 84672 = 48 * (6 * 7)2 85264 = (4 * (68 + 5))2 85293 = (2 * 9 - 5) * 38
85358 = (5 + 8) * (38 + 5) 86142 = 21 * (84 + 6) 86724 = (74 + 8) * 62 87381 = (87 / 8 - 1) / 3 87382 = (87 / 8 + 2) / 3
91125 = (9 * 5 * 1)2+1 91853 = (9 + 5) * 38 - 1 91854 = (9 + 5) * (4 - 1)8 92160 = 10 * 962 93184 = 91 * 48-3
93217 = 97 * 312 93294 = 2 * ((4 * 9)3 - 9) 93312 = 2 * (9 * (3 + 1))3 93642 = (9 * 34)2 + 6 94395 = 93 * (45 - 9)
95232 = 93 * 22*5 95234 = 93 * 45 + 2 97333 = (39 + 7)3 - 3 97336 = (39 + 7)6-3 97343 = (49 - 3)3 + 7
97375 = 779 * 53 97966 = 76 - (9 - 6)9 98256 = 6 * (29+5 - 8) 98304 = 3 * 89-4 + 0 98305 = 3 * 85 + 90
98325 = 3 * (85 + 9 - 2) 98375 = 5 * (9 * 37 - 8) 98415 = 98-4 * 15 (MR) 98435 = 5 * (39 + 8 - 4) 99225 = ((9 - 2) * 9 * 5)2

Here's a plot of the 837 Friedman numbers less than 10,000, where the x-coordinate is the rank of the number and the y-coordinate is the number. Thus flat places are concentrations of Friedman numbers and steep places are intervals with few Friedman numbers.

Philippe Fondanaiche says the smallest repdigit Friedman number appears to be 99999999. I improved some of his smallest repdigits, which are shown below.

11111111111 = ((11-1)11 - 1*1) / (11-1-1)
22222222222222 = (2((22-2)/2)22+2-2-2) / (2+2/2)2
333333333 = ((3*3 + 3/3)3*3 - 3/3) / 3
444444444444444 = (4(44/4 - 4/4)4*4-4/4 - 4) / (4 + 4 + 4/4)
5555555555 = (5(5+5)5+5 - 5) / (5 + 5 - 5/5)
6666666666666666 = (6((66-6)/6)6 + (66-6)/6 - 6) / (6 + (6+6+6)/6)
77777777777777 = (7((77-7)/7)7+7 - 7 + 7 - 7) / (7 + (7+7)/7)
88888888888888 = (8((88-8)/8)8+8-(8+8)/8 - 8) / (8 + 8/8)
99999999 = (9 + 9/9)9-9/9 - 9/9

Brendan Owen proved that repdigits of length 24 or more are Friedman numbers in any base, by showing:

aaa...a = (a*a / (aa-a-a) ) ( ( (aa-a)/a)A + (a+a+...+a)/a - a/a), where A = ((a+a+a+a+a)/a)(a+a)/a - a/a.

The following people proved that F(n)/n --/--> 0. Brendan Owen used N,12588304 = N * 108 + 35482 (along with dozens of larger examples). Mike Reid used N,46656 = N * (4+6)5 + 66. Note that these examples also show that there are Friedman numbers beginning with any string of digits.

Combining all the known examples of this type, the best bound is about liminf F(n)/n > .000011196. I still suspect that F(n)/n --> 1 despite the fact that there are arbitrarily large numbers (like powers of 10) that are not Friedman numbers.

Ulrich Schimke conjectures that for every k which is not a power of 10, kn is a Friedman number for arbitrarily large n. He notes that 2n appears to be a Friedman number for all n>9. Trevor Green points out that all powers of 5 seem to be Friedman numbers.

A vampire number is a number that can be written as the product of numbers that together contain the same digits as the number itself. This sequence, a subsequence of the Friedman numbers, begins 126, 153, 688, 1206, 1255, 1260, 1395, 1435, 1503, 1530, 1827, 2187, 3159, 3784, 6880, . . . , and is sequence A020342 of the Encyclopedia of Integer Sequences. Philippe Fondanaiche sent me lots of vampire numbers, but noticed that vampire numbers get more and more rare, so that the vast majority of Friedman numbers use the exponential operator. Trevor Green noticed that 12(n+1) and 1n3 are always vampire numbers in base 2n, and that they appear to always be the smallest such.

Philippe Fondanaiche noticed that most Friedman numbers are composite. The first few known prime Friedman numbers are 127, 347, 2503, 12101, 12107, 12109, 15629, 15641, 15661, 15667, 15679, 16381, 16447, 16759, 19739, . . . .

Ron Kaminsky proved that there are infinitely many prime Friedman numbers. The numbers k*1014+19683 = k*106+8+39+0+0+ . . . are Friedman numbers for all k. The numbers of this form are an arithmetic sequence a n+b where a and b are relatively prime, and therefore, by a well-known theorem of Dirichlet, the sequence contains an infinite number of primes.

Trevor Green proved that there are infinitely many Friedman numbers in every base by considering numbers of the form 1000...02000...01=1000...012+0+0+...+0 in bases larger than 2 and numbers of the form 1000...01000...0001=1000110+0+0+...+0 in base 2.

Trevor Green also writes: "25 is a Friedman number in bases 2, 3 and 4 as well as base 10. What other numbers are Friedman numbers in more than one base, or in an unusually large number of bases? What numbers are not Friedman numbers in any base?"

Trevor Green has found several other strings which are Friedman numbers in all large bases, such as 102030201 = (10301 - 200)2 and 1367631 = (117 - 6)(6+3)/3. Also, he found the series 121 = 112, 12321 = (113 - 2)2, 1234321 = (1143 - 32)2, 123454321 = (11543 - 432)2, . . . .

Here are the small Friedman numbers in other bases:

Friedman Numbers in Base 2
11001 = 10110 11011 = 110+11 111111 = (11 + 1)11 - 1 1001111 = 11100 - 1 - 1 1010001 = 11100 + 0 + 0
1010011 = 11100 + 10 1100011 = 101010 - 1 1100100 = 101010 + 0 1100101 = 101010 + 1 1111001 = 1 * 101110
1111010 = 101110 + 1 1111011 = 10111 - 1 - 1 1111100 = 10111 - 1 - 0 1111101 = 10111 + 1 - 1 1111110 = 10111 + 1 * 1
1111111 = (1 + 1)111 - 1 * 1

Friedman Numbers in Base 3
121 = 112 221 = 122 1022 = 202 - 1 1122 = 2 * 211 1211 = 211+1
1212 = 1 + 212 2022 = 220 - 2 2101 = (1 + 1)20 2102 = 220 + 1 10122 = 1012 - 2
10201 = 1012 + 0 11022 = 2 * 2011 11122 = 1 * 121+2 11202 = (1 + 1)20 * 2 11220 = 2 * 2110
12021 = 1102 - 2 12022 = -1 + (20 * 2)2 12100 = 1102 + 0 12101 = 1102 + 1 12102 = 1102 + 2
12122 = 2 * 2211 12221 = 12 * 212 21021 = 1122 + 0 21102 = 2 * 1012 22021 = 1202 - 2
22100 = 1202 + 0 22101 = 1202 + 1 22102 = 1202 + 2 100111 = (10 + 0 + 1)11 100112 = 2100-1 + 1
101122 = 2 * 1102 - 1 101201 = (200 - 1)1+1 101202 = (200 - 1)2 + 1 101222 = 11 * 2202 102202 = 2002 - 21
102212 = 1 * 12 * 220 102220 = 2002 - 2 - 1 102221 = (201 - 1)2 - 2 102222 = (202 / 2)2 - 1 110022 = 2 * 20011
110122 = 2110 - 2 * 1 110201 = 2110 - 1 * 0 110202 = 2110 - 20 110220 = 2 * 20110 111022 = 2012 - 1 - 1
111102 = 2011+1 + 1 111220 = (1 + 2) * 1210 112002 = 2 * 201001 112012 = 10 * 221 - 1 112020 = (1 + 1)20 * 20
112021 = 10 * 221 + 1 112022 = 10 * 221 + 2 112112 = 1121+1 * 2 112200 = 2 * 21100 112202 = 2022 - 1 - 1
112210 = 2021+1 - 1 112211 = (211 - 2)1+1 112212 = (211 - 2)2 + 1 112220 = 2022 + 1 + 1 120221 = 2 * (201+2 - 1)
121002 = 2 * (1020 + 1) 121012 = 2102 - 11 121020 = 2102 - 10 121021 = -1 + 2102 - 1 121022 = 1 + 2102 - 2
121100 = 2101+1 + 0 121101 = 2101+1 + 1 121102 = 2101+1 + 2 121121 = 21 * 111+2 121200 = 10102 / 2
121202 = 2 * (1202 + 1) 121220 = 2 * 22110 122012 = 22 * 2011 122122 = 2112 - 22 122201 = 2112 - 20
122210 = 120 * 212 122211 = 2112 - 2 - 1 122212 = 2 * ((1 + 2)12 + 2) 122220 = (202 + 2)2 - 1 122221 = (2 * (2 * 12 + 1))2
200221 = 2102 - 20 200222 = (2 + 0 + 0) * 222 201121 = 2121+1 + 0 201122 = 2122 + 1 + 0 210022 = 2202 - 1 - 0
211020 = 20 * 1012 211212 = 1211 - 22 211222 = 122*2 - 12 212011 = (21 - 2)0+11 212012 = 122*2 + 0 + 1
212021 = 122*2 + 10 212101 = 1211 + 20 212112 = 12 * (221 - 1) 212122 = 12 * 221 - 2 212201 = 12 * 221 + 0
220221 = 2222 - 10 220222 = (220 + 2)2 - 2 221001 = (1001 - 2)2 221021 = 2 * (2110 - 2) 221101 = 2 * 2110 - 1
221102 = 2 * 211+0+2 221110 = 2 * 2110 + 1 221120 = 2 * (2110 + 2) 221221 = 21 * (2 + 12)2 222200 = 202 * 202
222201 = (1 + 2)20 - 22 222202 = (202 + 2)2 / 2 222221 = (1 + 2)2+2+2 - 2

Friedman Numbers in Base 4
121 = 112 123 = (1 + 2)3 1203 = 3 * 201 1230 = 3 * 210 1321 = 231+1
1322 = 232 + 1 1331 = (3 + 1 + 1)3 1332 = (3 + 2)3 + 1 2032 = 302 - 2 2213 = 312 - 2
3120 = 123 + 0 3121 = 123 + 1 3122 = 123 + 2 3123 = 3 + 123 3322 = 2 * (3 * 2)3
10132 = 1012 - 3 10201 = 1012 + 0 10221 = 21 * 20111113 = 131+1+1 11133 = 3 * 1311
11221 = (111 - 2)2 11313 = 113 * 1 * 3 12003 = 3 * 2001 12030 = 3 * 2010 12031 = 1102 - 3
12100 = 1102 + 0 12101 = 1102 + 1 12102 = 1102 + 2 12103 = 1102 + 3 12232 = 2 * 123 - 2
12233 = -1 + 2 * (2 * 3)3 12300 = 3 * 2100 12303 = 3 * (302 + 1) 12321 = ((1 + 2) * 13)2 12323 = (3 * 13)2 + 2
13023 = 21 * 303 13211 = 2 * 311 + 1 13212 = 2 * 312-1 13222 = 132 * 22 13233 = 313 * 32
13322 = 23*3 - 12 13323 = (3 * 31)2 / 3 13331 = (1 + 1)3*3 - 3 13332 = (-1 + 3)3*3 - 2 21233 = (2 + 3)1+3 - 2
21301 = 113+20 21331 = 31 * 132 21333 = 3 * (123 - 3) 22210 = 1222 + 0 22211 = 1222 + 1
22212 = 1222 + 2 22213 = 1222 + 322330 = 32 * 302 22332 = 3 * 3222 23031 = 312 - 30
23102 = 2 * 1032 23112 = 312 - 1 - 2 23113 = -2 + 3(1+1)*3 23120 = 312 - 20 23121 = (23 + 1)2+1
23122 = 32+2+2 + 1 23123 = 2 + 31*2*3 23130 = 312 + 3 + 0 23131 = 312 + 3 + 1 23132 = 312 + 3 + 2
23133 = 312 + 3 + 3 23201 = 312 + 20 23213 = 312 + 32 23322 = 3 * (223 - 2) 31021 = 1312 + 0
31323 = 31+3 * 23 33211 = (21 + 1)3 - 3 33212 = (32 + 1)3 - 2 33220 = ((2 + 3) * 2)3 + 0 33221 = ((2 + 3) * 2)3 + 1
33222 = ((2 + 3) * 2)3 + 2 33223 = 3 + ((3 + 2) * 2)3 33232 = 223 + 3 + 3 33312 = (3 + 3) * 312 33322 = 2 * (23*3 - 3)

Friedman Numbers in Base 5
121 = 112 224 = 22+4 1232 = 22 * 31 1241 = 241+1 1242 = 1 + 242
1331 = (3 * (1 + 1))3 1332 = (3 * 2)3 + 1 1433 = 143 / 3 1443 = 44 - 13 2123 = 322 - 1
2124 = (24 + 1)2 2244 = (4 * 4 + 2)2 2333 = (-2 + 3 * 3)3 2421 = (41 - 2)2 2423 = 342 + 2
2433 = 4 * 332 3042 = 402 - 3 3421 = 2 * 34+1 3422 = 3 + 422 4243 = 442 - 3
4441 = (4 + 1)4 - 4 10142 = 1012 - 4 10201 = 1012 + 0 10314 = 311 - 40 10413 = 311 + 4 + 0
11424 = 44 * 112 12041 = 1102 - 4 12100 = 1102 + 0 12101 = 1102 + 1 12102 = 1102 + 2
12103 = 1102 + 3 12104 = 1102 + 4 12144 = 4 * 21 * 41 12320 = 22 * 130 12321 = (2 * 31 - 1)2
12324 = 2 * 3412 12340 = 4 * 310 - 2 12342 = 4 * (1 + 2)2+3 12343 = 4 * 32+3 + 1 12344 = 4 * 31+4 + 2
13041 = 1 * 410 - 3 13043 = 430/3 - 1 13044 = (1 + 3)0+4 * 4 13102 = (1 + 1)20 + 3 13321 = 1132 - 3
13323 = 32 * (1 + 3)3 13324 = (3 * (3 * 4 - 1))2 14111 = 1141+1 14112 = 1142 + 1 14214 = 4 * 2141
14330 = 1 * 30 * 34 20141 = (12 - 1)4 + 0 20311 = 1 * 213 + 0 20312 = 213 + 20 21144 = 4 * 2411
21232 = -2 + 1232 21234 = (1 * 2 * 34)2 21243 = 1232 + 4 21311 = 2 * (-1 + 311) 21313 = 2 * (13 + 1)3
21314 = 2 * 143 + 1 21414 = 24 * 411 22041 = (2 * 40 - 1)2 22314 = 21 * (3 * 4)2 22400 = (2 * 40)2 + 0
22401 = (2 * 40)2 + 1 22402 = 2 + (2 * 40)2 22403 = (2 * 40)2 + 3 22404 = (2 * 40)2 + 4 23211 = (132 - 1)2
23212 = 2132 / 2 23334 = (3 * 4)3 - 32 23341 = (3 * 4)3 - 12 23343 = -2 - 3 (3 * 4)3 23344 = (23 + 4)3 - 4
23402 = 223 - 40 23403 = (23 + 4 + 0)3 23412 = 223 + 1 * 4 23434 = (3 * 4)3 + 24 24024 = (40 + 42)2
24132 = 31 * 422 24244 = 424 * 24 24330 = 4 * 3320 24343 = 42 * (34 + 3) 24344 = (34 + 44)2
30234 = (40 - 2)3 / 3 30444 = (403 - 4) / 4 31042 = 1402 - 3 31134 = 4 * (133 - 1) 31142 = 23*4-1 - 1
31143 = (3 - 1)-1+4*3 31144 = 13 * 44 + 1 31204 = 2 * (410 + 3) 31422 = 221 + 43 32221 = 312 - 2 / 2
32222 = 3(22+2)/2 32224 = 324/2 + 2 32233 = 3 * (32*3 + 2) 32242 = (24 - 2 / 2)3 32314 = (134 + 3)2
33204 = (4 * (30 - 3))2 33222 = 23 * 322 34021 = 124 - 30 34041 = (3 + 4)4 - 10 34042 = (20 - 3)4 - 4
34043 = (3 + 4)0+4 - 3 34101 = (3 + 4)10-1 34102 = 124 + 30 34210 = 20 * 31+4 34212 = 34 * 212
34412 = 124 + 34 34423 = 4 * ((2 + 3)4 - 3) 40214 = 20 * 44 - 1 40332 = 2 * (3 + 3)4 + 0 41122 = 2 * 214-1
43124 = 4 * 143 - 2 43131 = 4 * 33*(1+1) 43132 = 4 * 33*2 + 1 43134 = 4 * 143 + 3 43234 = 43 * 23+4
44231 = 3 * (41+4 - 2) 44233 = -4 + 42+3 * 3 44242 = 44 * (2 + 4) * 2 44301 = 3 * 410 + 4 44313 = 413 / 3 - 4
44314 = 3 * 4 * (44 + 1) 44331 = 413 / 3 + 4

Friedman Numbers in Base 6
24 = 24 52 = 25 121 = 112 124 = 4 * 21 133 = 3 * 31
143 = -1 + 43 144 = 44-1 213 = 132 1043 = 34+1 + 0 1053 = 35 + 10
1135 = 5 * 131 1204 = 4 * 201 1224 = 2 * 4121240 = 4 * 210 1242 = 2 * 421
1252 = 5 * 122 1303 = 3 * 301 1330 = 3 * 310 1352 = 312 - 5 1353 = 3 * 315
1423 = 23 * 41 1425 = 21 * 45 1524 = 4 * 251 1533 = 3 * 351 2212 = 2(2+1)2
2213 = 232 + 1 2235 = 352 - 2 2241 = (41 - 2)2 2355 = 35 * 52 2400 = 402 + 0
2401 = 402 + 1 2402 = 2 + 402 2403 = 402 + 3 2404 = 402 + 4 2405 = 402 + 5
2514 = 54 - 2 - 1 2515 = -2 + 55-1 2521 = 1 * 52+2 2534 = 54 + 32 2535 = (2 + 53) * 5
2544 = 5 * 44 / 2 3213 = 32+1+3 3214 = 34+2 + 1 3215 = 35+1 + 2 3453 = 5 * 433
3513 = 51 * 33 3524 = 33 * 51 4052 = 502 - 4 4352 = 45 - 32 4415 = 44+1 - 5
4424 = 44+2 / 4 4431 = 44+1 + 3 4432 = 4 + 43+2 4435 = 45 + 4 + 3 4452 = 45 + 24
4504 = 45 + 40 4515 = 45 + 51 5204 = 542 + 0 5325 = 5 * (35 - 2) 5343 = 5 * 34 * 3
5432 = (54 - 3) * 2 5442 = 54 * 4 / 2 5532 = 5 * 25+3

Friedman Numbers in Base 7
121 = 112 143 = 34 - 1 144 = (-1 + 4)4 264 = 4 * 62 514 = (5 - 1)4
1155 = 15 * 51 1253 = 312 - 5 1263 = 2 * 36-1 1331 = (1 + 1)3*3 1332 = 23*3 + 1
1452 = 51 * 42 1541 = 54 - 11 1544 = -1 + 54 - 4 1545 = 1 + 54 - 5 1551 = -1 + 55-1
1552 = 1 * (5 * 5)2 1553 = 1 + 5 * 53 1614 = 64 / (1+1) 2061 = (2 + 1)6 + 0 2063 = 36 + 2 + 0
2314 = 412 + 3 2343 = 4 * (2 * 3)3 2422 = -2 + 422 2424 = 424-2 2534 = 432 + 5
2542 = (45 - 2)2 2635 = (5 * 2)3 + 6 2640 = 40 * 62 2654 = 26+4 - 5 2662 = 26+6-2
3354 = 5 * 3 * 34 3425 = 2 * (54 - 3) 3464 = -34 + 64 3531 = (3 + 3)5-1 4323 = 3 * (4 * 2)3
4625 = (64 - 5)2 5062 = 602 - 5 5446 = (65 + 4) / 4 5622 = (65 - 2)2 6243 = (6 / 2)4+3
6265 = 652 - 6

Friedman Numbers in Base 8
33 = 33 121 = 112 125 = 5 * 21 143 = 3 * 41 251 = 152
257 = 7 * 52 326 = 63 - 2 363 = 36 / 3 527 = 75-2 1133 = 3 * 311
1205 = 5 * 201 1250 = 5 * 210 1326 = 132 * 6 1327 = 37-1 - 2 1331 = 33*(1+1)
1332 = 1 + 33*2 1356 = 5 * 6 * 31 1403 = 3 * 401 1430 = 30 * 41 1626 = 21 * 66
1724 = 7 * 214 2147 = 7 * 241 2204 = 422 + 0 2342 = (2 + 3)4 * 2 2345 = 2 * 54 + 3
2346 = 3 * 642 2372 = 32 * 72 2416 = 64 - 1 * 2 2534 = (2 + 5)3 * 4 2544 = 5 * 424
2570 = 70 * 52 2642 = 462 - 2 2644 = 464-2 2662 = (2 / 6)-6 * 2 2754 = 472 - 5
3245 = 35 * 34 3275 = 5 * (73 + 2) 3534 = 3 * (54 + 3) 4213 = 34+2+1 4217 = 4 + (2 + 1)7
4237 = 37 + 24 4334 = 34 * 34 4527 = 74 - 5 * 2 4534 = (3 + 4)4 - 5 4537 = 74 - 5 + 3
4541 = (14 - 5)4 4572 = 74 + 52 4576 = 5 * 746 4654 = 4 * (-6 + 54) 5374 = 54 + 37
5676 = 6 * 765 5726 = 672 + 5 6065 = (-60 + 6)5 6072 = 702 - 6 7246 = (7 - 2)4 * 6

Friedman Numbers in Base 9
121 = 112 134 = 4 * 31 314 = (3 + 1)4 628 = 8 * 26 1304 = 4 * 301
1326 = 2 * 613 1340 = 31 * 40 1354 = 45 - 1 * 3 1357 = 1 * (7 - 3)5 1362 = 2 * 631
1363 = 3 * (1 + 6)3 1438 = 18 * 43 1456 = 4 * 6 * 51 2086 = 20+8 * 6 2132 = (32 - 1)2
2136 = 6 * 321 2467 = 472 - 6 2472 = -2 + 472 2474 = 474-2 2725 = 27+5 / 2
2737 = -27 + 37 3254 = -3 + (2 + 5)4 3247 = 74 - 32 3257 = 75+2-3 3454 = 5 * (4 + 4)3
3458 = 5 * 83 + 4 3672 = (2 * 7)3 - 6 3678 = (7 * (8 - 6))3 4126 = 612 - 4 4252 = (4 + 2 / 2)5
4357 = 37 + 45 5485 = 84 - 55 6280 = 80 * 26 6827 = 782 + 6 7082 = 802 - 7
8836 = 38 - 6 * 8 8873 = 38 - 7 - 8

Here are some other results of mine on Friedman numbers. There are arbitrarily long strings of consecutive Friedman numbers, because of the numbers from 25*102n to 25*102n + 10n - 1 are 10n consecutive Friedman numbers. For example, 250068 = 5002+68. This also shows that there are Friedman numbers ending with any string of digits.

It is also easy to show that if n>60, there is a Friedman number between n and 2n. Zeroes can be added to the right of any of the Friedman numbers 688 = 8 * 86, 1206 = 6 * 201, 1827 = 21 * 87, 3159 = 9 * 351, and 3784 = 8 * 473. The list at the top of the page handles the small cases.

If you can extend any of these results, please e-mail Erich Friedman.