cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A328277 Triangle T(m,n) = # { k | concat(mk,nk) has no digit twice or more }, m > n > 0.

Original entry on oeis.org

304, 153, 157, 197, 124, 97, 221, 156, 69, 171, 73, 88, 142, 68, 69, 129, 73, 81, 86, 62, 46, 189, 88, 40, 67, 48, 51, 24, 89, 80, 77, 31, 63, 68, 41, 20, 0, 132, 80, 90, 58, 32, 63, 99, 37, 0, 106, 69, 79, 50, 30, 45, 30, 38, 0, 76, 0, 96, 31, 62, 54, 27, 31, 49, 41, 27, 84, 72, 0, 31, 58, 47, 26, 23, 34, 43, 25, 20
Offset: 2

Views

Author

M. F. Hasler, Oct 10 2019

Keywords

Comments

Row m has columns numbered n = 1 .. m-1, with m >= 2.
For an extension to m >= n >= 0, see A328288, and A328287 for column 0.
One consider T(m,n) defined for all m, n >= 0, which would yield a symmetric, infinite square array T(m,n), see formula.
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.

Examples

			The table reads:
  304,     (m=2)
  153, 157,
  197, 124,  97,
  221, 156,  69, 171,
   73,  88, 142,  68, 69,
  129,  73,  81,  86, 62, 46,
  189,  88,  40,  67, 48, 51, 24,
   89,  80,  77,  31, 63, 68, 41, 20,
    0, 132,  80,  90, 58, 32, 63, 99, 37,
    0, 106,  69,  79, 50, 30, 45, 30, 38,  0,    (m = 11)
   76,   0,  96,  31, 62, 54, 27, 31, 49, 41, 27,
   84,  72,   0,  31, 58, 47, 26, 23, 34, 43, 25, 20,
  100,  64,  52,   0, 51, 44, 51, 42, 22, 38, 27, 18, 20
  ...
The terms corresponding to T(2,1) = 304 and T(3,1) = 153 are given in Eric Angelini's post to the SeqFan list.
T(8,7) = 24 = #{1, 5, 7, 9, 12, 51, 71, 76, 105, 107, 122, 128, 132, 134, 262, 627, 674, 853, 1172, 1188, 1282, 1321, 2622, 5244}: For these numbers k, 8k and 7k don't share any digit and have no digit twice; e.g., 5244*(8,7) = (41952, 36708).

Links

Crossrefs

Cf. A328288 (variant m >= n >= 0), A328287 (column 0).

Programs

  • PARI
    A328277(m,n)={my(S,s); for(L=1,10,S<(S+=sum( k=10^(L-1),10^L-1, #Set(Vecsmall(s=Str(m*k, n*k)))==#s))||L<3||return(S))} \\ Using concat(digits...) would take about 50% more time.

Formula

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.

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

Views

Author

M. F. Hasler, Oct 10 2019

Keywords

Comments

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.

Examples

			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).

Links

Programs

Formula

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

Views

Author

M. F. Hasler, Oct 11 2019

Keywords

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?

Examples

			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.

Crossrefs

Cf. A328287 (row 10), A328288, A328277.
Column 1 is A007526 (number of nonnull variations of n distinct objects).

Programs

  • 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}

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.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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