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

%I A328277 #22 Dec 23 2024 14:53:46
%S A328277 304,153,157,197,124,97,221,156,69,171,73,88,142,68,69,129,73,81,86,
%T A328277 62,46,189,88,40,67,48,51,24,89,80,77,31,63,68,41,20,0,132,80,90,58,
%U A328277 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
%N A328277 Triangle T(m,n) = # { k | concat(mk,nk) has no digit twice or more }, m > n > 0.
%C A328277 Row m has columns numbered n = 1 .. m-1, with m >= 2.
%C A328277 For an extension to m >= n >= 0, see A328288, and A328287 for column 0.
%C A328277 One consider T(m,n) defined for all m, n >= 0, which would yield a symmetric, infinite square array T(m,n), see formula.
%C A328277 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.
%H A328277 M. F. Hasler, in reply to E. Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2019-October">Fractions with no repeated digits</a>, SeqFan list, Oct. 10, 2020.
%F A328277 T(m,n) = 0 whenever m = n (mod 10).
%F A328277 T(m,n) = T(n,m) for all m, n >= 0, if the condition m > n is dropped.
%e A328277 The table reads:
%e A328277   304,     (m=2)
%e A328277   153, 157,
%e A328277   197, 124,  97,
%e A328277   221, 156,  69, 171,
%e A328277    73,  88, 142,  68, 69,
%e A328277   129,  73,  81,  86, 62, 46,
%e A328277   189,  88,  40,  67, 48, 51, 24,
%e A328277    89,  80,  77,  31, 63, 68, 41, 20,
%e A328277     0, 132,  80,  90, 58, 32, 63, 99, 37,
%e A328277     0, 106,  69,  79, 50, 30, 45, 30, 38,  0,    (m = 11)
%e A328277    76,   0,  96,  31, 62, 54, 27, 31, 49, 41, 27,
%e A328277    84,  72,   0,  31, 58, 47, 26, 23, 34, 43, 25, 20,
%e A328277   100,  64,  52,   0, 51, 44, 51, 42, 22, 38, 27, 18, 20
%e A328277   ...
%e A328277 The terms corresponding to T(2,1) = 304 and T(3,1) = 153 are given in Eric Angelini's post to the SeqFan list.
%e A328277 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).
%o A328277 (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.
%Y A328277 Cf. A328288 (variant m >= n >= 0), A328287 (column 0).
%K A328277 nonn,base,tabl,fini
%O A328277 2,1
%A A328277 _M. F. Hasler_, Oct 10 2019