A217368 Smallest number having a power that in decimal has exactly n copies of all ten digits.

32043, 69636, 643905, 421359, 320127, 3976581, 47745831, 15763347, 31064268, 44626422, 248967789, 85810806, 458764971, 500282265, 2068553967, 711974055, 2652652791, 901992825, 175536645, 3048377607, 3322858521, 1427472867, 3730866429, 9793730157

Offset: 1

Author

James G. Merickel, Oct 01 2012

Keywords

• nonn
• base

Comments

The exponents that produce the number with a fixed number of copies of each digit are listed in sequence A217378. See there for further comments.

Since we allow A217378(n)=1, the sequence is well defined, with the upper bound a(n) <= 100...99 ~ 10^(10n-1) (n copies of each digit, sorted in increasing order, except for one "1" permuted to the first position). - M. F. Hasler, Oct 05 2012

What is the minimum value of a(n)? Can it be proved that a(n) > 2 for all n? - Charles R Greathouse IV, Oct 16 2012

Examples

			The third term raised to the fifth power (A217378(3)=5), 643905^5 = 110690152879433875483274690625, has three copies of each digit (in its decimal representation), and no number smaller than 643905 has a power with this feature.
		

Crossrefs

Cf. A020666, A054038, A054039, A066825, A067635, A074205, A156977, A217378.

Programs

Mathematica

f[n_] := Block[{k = 2, t = Table[n, {10}], r = Range[0, 9]}, While[c = Count[ IntegerDigits[k^Floor[ Log[k, 10^(10 n)]]], #] & /@ r; c != t, k++]; k] (* Robert G. Wilson v, Nov 28 2012 *)

PARI

is(n,k)=my(v);for(e=ceil((10*n-1)*log(10)/log(k)), 10*n*log(10)/log(k), v=vecsort(digits(k^e)); for(i=1,9,if(v[i*n]!=i-1 || v[i*n+1]!=i, return(0))); return(1)); 0
a(n)=my(k=2); while(!is(n,k),k++); k \\ Charles R Greathouse IV, Oct 16 2012

Extensions

a(13)-a(14) from James G. Merickel, Oct 06 2012 and Oct 08 2012

a(15)-a(16) from Charles R Greathouse IV, Oct 17 2012

a(17)-a(19) from Charles R Greathouse IV, Oct 18 2012

a(20) from Charles R Greathouse IV, Oct 22 2012

a(21)-a(24) from Giovanni Resta, May 05 2017