A328277 -id:A328277 - cp's OEIS frontend

A328287 Number of multiples of n which have only distinct and nonzero digits in base 10.

0, 986409, 438404, 572175, 219202, 109601, 255752, 140515, 109601, 432645, 0, 90212, 127163, 75768, 62436, 65027, 56104, 57930, 194244, 51869, 0, 81493, 40572, 42969, 63654, 27400, 33587, 145926, 31217, 34146, 0, 31827, 27926, 51090, 25772, 15702, 97114, 26330, 23106, 43929, 0, 23983, 36409

Offset: 0

M. F. Hasler, Oct 10 2019

This is column 0 of A328288, extension of A328277.
No term can exceed a(1), cf. example. The sequence is finite in the sense that a(n) = 0 for n > 987654321, since one cannot have more than 9 distinct nonzero digits.
See A328290 for generalization to other bases.

			For n = 1, this is simply the number of numbers with only distinct and nonzero digits. All other terms are less than a(1), namely, the size of the subset of these numbers which are multiples of n.

Rémy Sigrist, PARI program for A328287

Cf. A328288, A328277.

This is row 10 of A328290.

A328287(n,B=10,S)={for(L=1,B-1,my(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} \\ 2nd optional argument allows to specify a base different from 10

PARI
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See Links section
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a(n) = 0 whenever n == 0 (mod 10) or n > 987654321.

A328288 Triangle T(m,n) = # { k | concat(mk,nk) has no digit twice or more }, m >= n >= 0.

Original entry on oeis.org

0, 986409, 0, 438404, 304, 0, 572175, 153, 157, 0, 219202, 197, 124, 97, 0, 109601, 221, 156, 69, 171, 0, 255752, 73, 88, 142, 68, 69, 0, 140515, 129, 73, 81, 86, 62, 46, 0, 109601, 189, 88, 40, 67, 48, 51, 24, 0, 432645, 89, 80, 77, 31, 63, 68, 41, 20, 0, 0, 0, 132, 80, 90, 58, 32, 63, 99, 37, 0

Offset: 0

M. F. Hasler, Oct 10 2019

This is an extended version of A328277 which is restricted to m > n >= 1.
One may consider T(m,n) defined for all m, n >= 0, which would yield a symmetric, infinite square array T(m,n), see formula.
For m and/or n = 0, see A328287(n) = T(0,n) = T(n,0), n >= 0.
The table is finite in the sense that T(m,n) = 0 for m > 987654321 (even if the multiple isn't pandigital, (mk, nk) cannot have more than 9+1 distinct digits), but also whenever the total number of digits of m and n exceeds 10.

			The table reads :
       0,    (m=0)
  986409, 0,      (m=1)
  438404, 304,   0,    (m=2)
  572175, 153, 157,   0,    (m=3)
  219202, 197, 124,  97,   0,   (m=4)
  109601, 221, 156,  69, 171,  0,   (m=5)
  255752,  73,  88, 142,  68, 69,  0,   (m=6)
  140515, 129,  73,  81,  86, 62, 46,  0,   (m=7)
  109601, 189,  88,  40,  67, 48, 51, 24,  0,   (m=8)
  432645,  89,  80,  77,  31, 63, 68, 41, 20,  0,   (m=9)
       0,   0, 132,  80,  90, 58, 32, 63, 99, 37,  0,   (m=10)
   90212,   0, 106,  69,  79, 50, 30, 45, 30, 38,  0,  0,    (m=11)
  127163,  76,   0,  96,  31, 62, 54, 27, 31, 49, 41, 27,   0,   (m=12)
   75768,  84,  72,   0,  31, 58, 47, 26, 23, 34, 43, 25, 20,   0,  (m=13)
   62436, 100,  64,  52,   0, 51, 44, 51, 42, 22, 38, 27, 18, 20   0,  (m=14)
  ...
The terms corresponding to T(2,1) = 304 and T(3,1) = 153 are given in Eric Angelini's post to the SeqFan list.
Column 0 is A328287 (number of multiples of m that have only distinct and nonzero digits).

M. F. Hasler, in reply to E. Angelini, Fractions with no repeated digits, SeqFan list, Oct. 10, 2020.

T(m,n)=if(min(m,n), A328277(m,n), A328287(max(m,n)))

T(m,n) = 0 whenever m == n (mod 10).

T(m,n) = T(n,m) for all m, n >= 0, if the condition m > n is dropped.

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

Original entry on oeis.org

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

M. F. Hasler, Oct 11 2019

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?

			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.

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

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

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}

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.

cp's OEIS Frontend

A328287 Number of multiples of n which have only distinct and nonzero digits in base 10.

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A328288 Triangle T(m,n) = # { k | concat(mk,nk) has no digit twice or more }, m >= n >= 0.

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A328290 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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