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%I #53 Feb 20 2024 07:01:31
%S 25,121,471663,931225,4473225,6953931,7301441,10713728,13246317,
%T 17332133,19367424,34706961,36310761,54363717,68714219,73553125,
%U 73641071,74390183,93478133,102712448,102941361,109502361,113162997,115521875,120934784,134179011,134381673,134472875,135478125
%N Numbers k whose prime factorization contains the same digits as k.
%C From _Robert G. Wilson v_, Jun 06 2014, updated Jun 10 2014: (Start)
%C The number of terms < 10^n: 0, 1, 2, 2, 2, 4, 7, 19, 71, 289, ..., .
%C There are only two terms which have just one prime factor (excluding multiplicity), i.e., 25 and 121. By index, they are 1 and 2.
%C The least term with just two prime factors is 471663. By index, they are 3, 4, 6, 7, 8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, ..., .
%C The least term with just three prime factors is 4473225. By index, they are 5, 9, 10, 11, 23, 24, 26, 28, 29, 30, 32, 36, 38, 39, 44, 46, 47, 66, ..., .
%C The least term with just four prime factors is 1713131455. By index, they are 110, 115, 251, ..., .
%C The least term with k prime factors (including multiplicity), or 0 if impossible or -1 not yet found, are 0, 25, 0, 931225, 7301441, 73553125, 471663, 4473225, 141294375, 251317472, 134179011, 1931229184, -1, 19367424, ..., .
%C So far ( < 10000000000) the count of digits 1,2,...,9,0 is {520, 271, 388, 254, 216, 211, 371, 172, 262, 117}.
%C (End)
%H Giovanni Resta, <a href="/A075047/b075047.txt">Table of n, a(n) for n = 1..1000</a> (first 311 terms from Robert G. Wilson v)
%e 25 = 5^2 and 121 = 11^2 are terms.
%e The term 1971753273 -> 1,9,7,1,7,5,3,2,7,3 -> 1,1,2,3,3,5,7,7,7,9 is in the sequence because its factorization is 3^7*7^1*37^1*59^2 -> 3,7,7,1,3,7,1,5,9,2 -> 1,1,2,3,3,5,7,7,7,9 and this coincides with the digits of the term itself. - _Robert G. Wilson v_, Jun 06 2014
%t fQ[n_] := Sort@ IntegerDigits@ n == Sort@ Flatten@ IntegerDigits@ FactorInteger@ n; k = 1; lst = {}; While[k < 100000001, If[ fQ@ k, AppendTo[lst, k]; Print@ k]; k++]; lst (* _Robert G. Wilson v_, Jun 05 2014 *)
%o (PARI) isok(n, b=10) = {f = factor(n); v = []; for (i=1, #f~, v = concat(v, digits(f[i,1], b)); v = concat(v, digits(f[i,2], b));); vecsort(v) == vecsort(digits(n, b));} \\ _Michel Marcus_, Jul 14 2015
%Y Cf. A025283, A036057, A064818.
%K base,nonn
%O 1,1
%A _Amarnath Murthy_, Sep 03 2002
%E More terms from _David Wasserman_, Jan 02 2005
%E a(14)-a(23) from _Donovan Johnson_, Oct 10 2009
%E a(24)-a(29) from _Robert G. Wilson v_, Jun 06 2014
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