A328290 Table T(b,n) = #{ k > 0 | nk has only distinct and nonzero digits in base b }, b >= 2, 1 <= n <= A051846(b).

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

Offset

2,2

Comments

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?

Links

Table of n, a(n) for n=2..120.

Formula

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.

Example

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.

Prog

(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}

Crossrefs

Cf. A328287 (row 10), A328288, A328277.

Column 1 is A007526 (number of nonnull variations of n distinct objects).

Sequence in context: A036875 A036877 A049763 * A182878 A221971 A378008

Adjacent sequences: A328287 A328288 A328289 * A328291 A328292 A328293

Keyword

nonn,base,tabf

Author

M. F. Hasler, Oct 11 2019

Status

approved