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A363160
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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.
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0
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1, 406512, 516473892, 5702631489, 961527834, 7025869314, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1
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OFFSET
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1,2
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COMMENTS
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For 7 <= n <= 185, I tried all possibilities with at most 10 distinct digits and I found no solution.
9876543210^186 has only 1859 < 186 * 10 = 1860 digits, so a(n) = -1 for n = 186.
So 9876543210^n has fewer than 10*n digits for n >= 186, so a(n) = -1 for n >= 186.
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LINKS
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EXAMPLE
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a(1) = 1, because 1^1 = 1 has each digit of 1, 1 time, and no lesser number > 0 satisfies this.
a(2) = 406512, because 406512 has distinct digits, 406512^2 = 165252006144 has each digit of 406512, 2 times, and no lesser number satisfies this.
n a(n) a(n)^n
1 1 1
2 406512 165252006144
3 516473892 137766973511455269432948288
4 5702631489 1057550783692741389295697108242363408641
5 961527834 821881685441327565743977956591832631269739424
6 7025869314 120281934463386157260042215510596389732740014997586987548736
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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sign,base
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AUTHOR
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STATUS
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approved
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