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%I #106 Sep 10 2023 09:09:35
%S 1,406512,516473892,5702631489,961527834,7025869314,-1,-1,-1,-1,-1,-1,
%T -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,
%U -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1
%N Smallest positive integer m with all digits distinct such that m^n contains each digit of m exactly n times, or -1 if no such m exists.
%C For 7 <= n <= 185, I tried all possibilities with at most 10 distinct digits and I found no solution.
%C 9876543210^186 has only 1859 < 186 * 10 = 1860 digits, so a(n) = -1 for n = 186.
%C So 9876543210^n has fewer than 10*n digits for n >= 186, so a(n) = -1 for n >= 186.
%e a(1) = 1, because 1^1 = 1 has each digit of 1, 1 time, and no lesser number > 0 satisfies this.
%e a(2) = 406512, because 406512 has distinct digits, 406512^2 = 165252006144 has each digit of 406512, 2 times, and no lesser number satisfies this.
%e n a(n) a(n)^n
%e 1 1 1
%e 2 406512 165252006144
%e 3 516473892 137766973511455269432948288
%e 4 5702631489 1057550783692741389295697108242363408641
%e 5 961527834 821881685441327565743977956591832631269739424
%e 6 7025869314 120281934463386157260042215510596389732740014997586987548736
%t hasDistinctDigitsQ[m_Integer?NonNegative]:=Length@IntegerDigits@m==Length@DeleteDuplicates@IntegerDigits@m;validNumberQ[n_Integer?NonNegative,m_Integer?NonNegative]:=AllTrue[Tally@IntegerDigits@m,Function[{digitFreq},MemberQ[Tally@IntegerDigits[m^n],{digitFreq[[1]],n*digitFreq[[2]]}]]];a[n_Integer?Positive,ex_Integer?Positive]:=Module[{m=1},Monitor[While[True,If[hasDistinctDigitsQ[m]&&validNumberQ[n,m],Return[m]];m++;If[m>10^(ex*n),Return[-1]];];m,m]];Table[a[n,7],{n,1,7}] (* _Robert P. P. McKone_, Sep 09 2023 *)
%Y Cf. A114258, A114259, A114260, A114261, A199630, A199631, A199632, A199633, A365144, A114260.
%K sign,base
%O 1,2
%A _Jean-Marc Rebert_, Sep 07 2023
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