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

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

Author

M. F. Hasler, Oct 10 2019

Keywords

• nonn
• base
• fini
• tabl

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

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

Programs

PARI

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

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.