A075047 Numbers k whose prime factorization contains the same digits as k.

25, 121, 471663, 931225, 4473225, 6953931, 7301441, 10713728, 13246317, 17332133, 19367424, 34706961, 36310761, 54363717, 68714219, 73553125, 73641071, 74390183, 93478133, 102712448, 102941361, 109502361, 113162997, 115521875, 120934784, 134179011, 134381673, 134472875, 135478125

Offset: 1

Author

Amarnath Murthy, Sep 03 2002

Keywords

• base
• nonn

Comments

From Robert G. Wilson v, Jun 06 2014, updated Jun 10 2014: (Start)

The number of terms < 10^n: 0, 1, 2, 2, 2, 4, 7, 19, 71, 289, ..., .

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.

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, ..., .

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, ..., .

The least term with just four prime factors is 1713131455. By index, they are 110, 115, 251, ..., .

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, ..., .

So far ( < 10000000000) the count of digits 1,2,...,9,0 is {520, 271, 388, 254, 216, 211, 371, 172, 262, 117}.

(End)

Examples

			25 = 5^2 and 121 = 11^2 are terms.
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
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 311 terms from Robert G. Wilson v)

Crossrefs

Cf. A025283, A036057, A064818.

Programs

Mathematica

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 *)

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

Extensions

More terms from David Wasserman, Jan 02 2005

a(14)-a(23) from Donovan Johnson, Oct 10 2009

a(24)-a(29) from Robert G. Wilson v, Jun 06 2014