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A328290
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Table T(b,n) = #{ k > 0 | nk has only distinct and nonzero digits in base b }, b >= 2, 1 <= n <= A051846(b).
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1
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1, 4, 1, 0, 0, 1, 0, 1, 15, 5, 9, 0, 2, 3, 2, 0, 4, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 64, 42, 21, 9, 0, 14, 8, 4, 7, 0, 6, 4, 3, 6, 0, 3, 2, 5, 2, 0, 4, 5, 3, 2, 0, 0, 2, 1, 2, 0, 2, 1, 2, 1, 0, 1, 2, 1, 2, 0, 1, 3, 1, 1, 0, 1, 1, 2, 1, 0, 0, 0, 0, 2
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OFFSET
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2,2
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COMMENTS
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The table could also be considered as an infinite square array with T(b,n) = 0 for n > A051846(b) = the largest pandigital number in base b.
Can anyone find a simple formula for the index of the last terms > 1 in each row b?
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LINKS
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FORMULA
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T(b,b) = 0, since any multiple of b has a trailing digit 0 in base b.
T(b,A051846(b)) = 1 and T(b,n) = 0 for n > A051846(b) = (b-1)(b-2)..21 in base b.
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EXAMPLE
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The table reads: (column n >= 2 corresponds to the base)
B \ n = 1 2 3 4 5 6 7 8 9 10 ...
2 1 (0 ...)
3 4 1 0 0 1 0 1 (0 ...)
4 15 5 9 0 2 3 2 0 4 1 ...
5 64 42 21 9 0 14 8 4 7 0 ...
6 325 130 65 65 161 0 48 23 32 66 ...
7 1956 651 1140 319 386 221 0 156 362 128 ...
8 13699 5871 4506 1957 2748 1944 6277 0 1470 1189 ...
9 109600 73588 27400 56930 21973 18397 15641 8305 0 14826 ...
10 986409 438404 572175 219202 109601 255752 140515 109601 432645 0 ...
(...)
In base 2, 1 is the only number with distinct nonzero digits, so T(2,1) = 1, T(2,n) = 0 for n > 1.
In base 3, {1, 2, 12_3 = 5, 21_3 = 7} are the only numbers with distinct nonzero digits, so T(3,1) = 4, T(3,2) = T(3,7) = T(3,7) = 1, T(3,n) = 0 for n > 7.
In base 4, {1, 2, 3, 12_4 = 6, 13_4 = 7, 21_4 = 9, ..., 321_4 = 57} are the only numbers with distinct nonzero digits, so T(4,n) = 0 for n > 57.
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PROG
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(PARI) T(B, n)={my(S, T, U); for(L=1, B-1, T=vectorv(L, k, B^(k-1)); forperm(L, p, U=vecextract(T, p); forvec(D=vector(L, i, [1, B-1]), D*U%n||S++, 2))); S}
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CROSSREFS
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Column 1 is A007526 (number of nonnull variations of n distinct objects).
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KEYWORD
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nonn,base,tabf
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AUTHOR
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STATUS
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approved
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