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%I #40 May 05 2017 12:37:51
%S 32043,69636,643905,421359,320127,3976581,47745831,15763347,31064268,
%T 44626422,248967789,85810806,458764971,500282265,2068553967,711974055,
%U 2652652791,901992825,175536645,3048377607,3322858521,1427472867,3730866429,9793730157
%N Smallest number having a power that in decimal has exactly n copies of all ten digits.
%C 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.
%C 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
%C 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
%e 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.
%t 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 *)
%o (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
%o a(n)=my(k=2); while(!is(n,k),k++); k \\ _Charles R Greathouse IV_, Oct 16 2012
%Y Cf. A020666, A054038, A054039, A066825, A067635, A074205, A156977, A217378.
%K nonn,base
%O 1,1
%A _James G. Merickel_, Oct 01 2012
%E a(13)-a(14) from _James G. Merickel_, Oct 06 2012 and Oct 08 2012
%E a(15)-a(16) from _Charles R Greathouse IV_, Oct 17 2012
%E a(17)-a(19) from _Charles R Greathouse IV_, Oct 18 2012
%E a(20) from _Charles R Greathouse IV_, Oct 22 2012
%E a(21)-a(24) from _Giovanni Resta_, May 05 2017
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