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

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

Author

M. F. Hasler, Oct 10 2019

Keywords

• nonn
• base
• tabl
• fini

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

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

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.