A363160 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.

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

Offset: 1

Author

Jean-Marc Rebert, Sep 07 2023

Keywords

• sign
• base

Comments

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.

Examples

			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
		

Crossrefs

Cf. A114258, A114259, A114260, A114261, A199630, A199631, A199632, A199633, A365144, A114260.

Programs

Mathematica

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 *)