Search a number
Friedman numbers
A number  $n$  is a Friedman number if it can be obtained combining all its digits with the 5 arithmetic operations and concatenation (of digits, not of results).

For example,  $13125$  is a Friedman number since it can be written as  $21\cdot5^{3+1}$.

M. Brand has proved that Friedman numbers have density 1, i.e., even if there are infinite numbers which are not Friedman (like all powers of 10), the probability that a number  $n$  is not Friedman asymptotically vanishes as  $n$  increases.

Many nice and interesting results about Friedman numbers are reported on the Erich Friedman web page linked below.

A Friedman number can be a repdigit, like

\[999999999= ((9 + 9 + 9)^{9 - 9}+9)^9 - 9/9\]
or a pandigital number, like
\[9108432576 = 251^3\cdot4\cdot6\cdot(7 + 8 + 9 + 0).\]

You can download a text file (Friedman_numbers_1e6.txt) containing a list of Friedman numbers up to  $10^6$  and their decompositions.

The first Friedman numbers 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 more terms

Friedman numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 83357 216023 551423 + 937537 ABA 128 1024 2048 + 968832 aban 25 121 125 + 736 abundant 126 216 736 + 999964 Achilles 2592 10368 15125 + 995328 admirable 3378 12102 12964 + 983046 alternating 25 121 125 + 983250 amenable 25 121 125 + 999964 apocalyptic 1285 1435 1503 + 29929 arithmetic 125 126 127 + 999964 astonishing 216 23490 automorphic 25 625 balanced p. 19739 74897 128153 + 885727 betrothed 186615 binomial 126 153 45760 + 857395 brilliant 25 121 289 + 966289 c.heptagonal 736 31256 46691 + 512072 c.nonagonal 293761 419986 597871 857395 c.octagonal 25 121 289 + 990025 c.pentagonal 273076 328516 387106 746656 c.square 25 3281 16381 + 683281 c.triangular 16381 24385 54436 + 457885 cake 2048 12384 121576 + 383439 Canada 125 Carmichael 46657 Chen 127 347 12107 + 995341 congruent 125 126 127 + 999964 constructible 128 1024 1285 + 983040 cube 125 216 343 + 970299 Cullen 25 Cunningham 126 127 2501 + 978122 Curzon 125 153 1530 + 995346 cyclic 127 347 1285 + 999163 D-number 15567 15627 16347 + 995331 d-powerful 153 1255 2048 + 995346 de Polignac 127 2503 12595 + 979769 decagonal 126 10251 32851 + 943326 deceptive 46657 216001 237169 + 754369 deficient 25 121 125 + 999163 dig.balanced 153 216 625 + 995463 droll 39424 93184 373248 + 565248 Duffinian 25 121 125 + 999163 eban 64036 economical 25 121 125 + 996543 emirp 347 12107 12109 + 976553 emirpimes 289 11663 12091 + 999163 enlightened 2048 2500 23328 + 262144 equidigital 25 121 127 + 996543 eRAP 263145 265225 616225 esthetic 121 343 12101 + 765432 Eulerian 524268 evil 125 126 153 + 999163 fibodiv 15612 54642 124896 Fibonacci 46368 frugal 125 128 343 + 995328 gapful 121 1260 1296 + 992256 good prime 127 347 16879 + 937571 happy 736 1285 4096 + 999964 Harshad 126 153 216 + 995364 heptagonal 13950 45360 59059 + 589761 hex 127 32761 166381 902557 hexagonal 153 46665 327645 + 857395 highly composite 1260 45360 498960 hoax 1255 2500 10192 + 983445 Hogben 343 326613 453603 Honaker 58921 64513 78163 + 902563 house 1285 iban 121 127 343 + 774144 idoneal 25 impolite 128 1024 2048 + 524288 inconsummate 216 2592 3281 + 996543 interprime 625 736 1827 + 995373 Jacobsthal 21845 43691 87381 Jordan-Polya 128 216 1024 + 995328 junction 216 1022 1024 + 996543 katadrome 765432 976521 976532 976542 Lehmer 1285 2509 21845 + 597871 Leyland 93312 262468 533169 Lucas 103682 lucky 25 127 289 + 995347 Lynch-Bell 126 128 216 + 912384 magic 125055 256040 265761 + 976625 magnanimous 25 625 736 metadrome 25 125 126 + 235678 modest 1022 1827 4088 + 640264 Moran 153 1503 12102 + 976572 Motzkin 127 narcissistic 153 nialpdrome 54432 65531 65542 + 999964 nonagonal 139500 291601 293625 978121 nude 126 128 216 + 983664 oban 25 625 688 736 octagonal 736 32865 87381 + 287680 odious 25 121 127 + 999964 Ormiston 250091 559813 773879 palindromic 121 343 10201 + 825528 pancake 121 4096 12091 + 857396 panconsummate 121 127 pandigital 216 1022 10575 + 781263 partition 1255 pentagonal 2501 12105 32782 + 399126 pernicious 25 121 127 + 997375 Pierpont 139969 147457 209953 + 786433 plaindrome 25 125 126 + 455679 Poulet 41665 46657 power 25 121 125 + 992016 powerful 25 121 125 + 995328 practical 126 128 216 + 995364 prim.abundant 3378 12102 12104 + 983046 prime 127 347 2503 + 995347 pronic 1260 86142 132860 + 792990 Proth 25 289 6145 + 983041 pseudoperfect 126 216 736 + 999964 repunit 121 127 343 + 597871 Rhonda 5832 15625 15698 + 342225 Ruth-Aaron 25 125 126 + 838936 Saint-Exupery 12960 103680 245760 + 937500 self 121 2503 3864 + 995346 semiprime 25 121 289 + 999163 sliding 25 2500 2504 + 393185 Smith 121 1255 3864 + 983445 sphenic 1022 1435 2505 + 983485 square 25 121 289 + 992016 star 121 226981 231673 + 683437 straight-line 23456 234567 765432 strong prime 127 347 2503 + 976559 super Niven 12600 126000 153000 282240 super-d 127 1792 3281 + 995331 superabundant 1260 tau 128 625 1260 + 995364 taxicab 13832 32832 216027 + 886464 tetrahedral 45760 triangular 153 46665 94395 + 857395 trimorphic 25 125 625 + 781249 truncatable prime 347 132647 276673 + 912647 twin 347 12107 12109 + 995341 uban 25 Ulam 126 688 1296 + 997246 undulating 121 343 unprimeable 625 1260 1792 + 992256 untouchable 216 1296 2048 + 997246 upside-down 8192 183729 261948 + 786423 vampire 1260 1395 1435 + 939658 wasteful 126 153 216 + 999964 weak prime 12101 12109 15629 + 995347 weird 125090 194390 227570 + 972268 Woodall 15624 114687 177146 + 524287 Zuckerman 128 216 1296 + 941184 Zumkeller 126 216 736 + 98304 zygodrome 116655 117755 117777 + 449955