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A090622 Square array read by antidiagonals of highest power of k dividing n! (with n,k>1).

%I A090622 #22 Jan 19 2017 16:27:54
%S A090622 1,0,1,0,1,3,0,0,1,3,0,0,1,1,4,0,1,0,1,2,4,0,0,1,1,2,2,7,0,0,0,1,1,2,
%T A090622 2,7,0,0,1,0,2,1,3,4,8,0,0,0,1,0,2,1,3,4,8,0,0,0,0,1,1,2,1,4,4,10,0,0,
%U A090622 0,1,1,1,1,4,2,4,5,10,0,0,1,0,1,1,2,1,4,2,5,5,11,0,0,0,1,0,1,1,2,1,4,2,5,5,11
%N A090622 Square array read by antidiagonals of highest power of k dividing n! (with n,k>1).
%H A090622 Alois P. Heinz, <a href="/A090622/b090622.txt">Antidiagonals n = 2..142, flattened</a>
%F A090622 For k=p prime: T(n,p) = [n/p] + [n/p^2] + [n/p^3] + .... For k = p^m a prime power: T(n,p^m) = [T(n,p)/m]. For k = b*c with b and c coprime: T(n,a*b) = min(T(n,a), T(n,b)). T(n,k) is close to, but below, n/A090624(k).
%e A090622 Square array starts:
%e A090622 1, 0, 0, 0, 0, 0, 0, ...
%e A090622 1, 1, 0, 0, 1, 0, 0, ...
%e A090622 3, 1, 1, 0, 1, 0, 1, ...
%e A090622 3, 1, 1, 1, 1, 0, 1, ...
%e A090622 4, 2, 2, 1, 2, 0, 1, ...
%e A090622 4, 2, 2, 1, 2, 1, 1, ...
%e A090622 7, 2, 3, 1, 2, 1, 2, ...
%p A090622 f:= proc(n, p) local c, k; c, k:= 0, p;
%p A090622        while n>=k do c:= c+iquo(n, k); k:= k*p od; c
%p A090622     end:
%p A090622 T:= (n, k)-> min(seq(iquo(f(n, i[1]), i[2]), i=ifactors(k)[2])):
%p A090622 seq(seq(T(n, 2+d-n), n=2..d), d=2..20);  # _Alois P. Heinz_, Oct 04 2012
%t A090622 f[n_, p_] := Module[{c = 0, k = p}, While[n >= k , c = c + Quotient[n, k]; k = k*p ]; c ]; t[n_, k_] := Min[ Table[ Quotient[f[n, i[[1]]], i[[2]]], {i, FactorInteger[k]}]]; Table[ Table[t[n, 2 + d - n], {n, 2, d}], {d, 2, 20}] // Flatten (* _Jean-François Alcover_, Oct 03 2013, translated from Alois P. Heinz's Maple program *)
%Y A090622 Columns include A011371, A054861, A090616, A027868, A054861, A054896, A090617, A090618, A027868, A064458, A090619, A090620, A054896, A027868, A090621.
%Y A090622 Cf. A090623, A090624, A115627.
%Y A090622 Diagonal gives A011776. - _Alois P. Heinz_, Oct 04 2012
%K A090622 nonn,tabl
%O A090622 2,6
%A A090622 _Henry Bottomley_, Dec 06 2003