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A202385 Number of partitions of n into distinct parts having pairwise common factors but no overall common factor.

%I A202385 #18 Feb 15 2017 07:55:11
%S A202385 1,0,0,0,0,0,1,0,0,0,2,0,2,0,0,0,2,0,2,0,1,0,4,0,3,0,1,0,5,0,8,0,2,0,
%T A202385 5,0,10,0,4,0,13,0,15,0,3,1,13,0,19,0,9,1,24,0,20,2,13,2,29,0,34,2,17,
%U A202385 2,34,1,49,2,21,3,58,2,63,3,20,7,72,2,81,3
%N A202385 Number of partitions of n into distinct parts having pairwise common factors but no overall common factor.
%H A202385 Alois P. Heinz, <a href="/A202385/b202385.txt">Table of n, a(n) for n = 31..251</a>
%e A202385 a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
%e A202385 a(37) = 1: [10,12,15] = [2*5,2*2*3,3*5].
%e A202385 a(41) = 2: [6,15,20], [6,14,21].
%e A202385 a(43) = 2: [6,10,12,15], [10,15,18].
%e A202385 a(53) = 4: [6,12,15,20], [15,18,20], [6,12,14,21], [14,18,21].
%e A202385 a(55) = 3: [10,12,15,18], [6,10,15,24], [6,21,28].
%p A202385 with(numtheory):
%p A202385 w:= (m, h)-> mul(`if`(j>=h, 1, j), j=factorset(m)):
%p A202385 b:= proc(n, i, g, s) option remember; local j, ok;
%p A202385       if n<0 then 0
%p A202385     elif n=0 then `if`(g>1, 0, 1)
%p A202385     elif i<2 then 0
%p A202385     else ok:= evalb(i<=n);
%p A202385          for j in s while ok do ok:= igcd(i, j)>1 od;
%p A202385          b(n, i-1, g, map(x->w(x, i), s)) +`if`(ok,
%p A202385          b(n-i, i-1, igcd(i, g), map(x->w(x, i), {s[], i}) ), 0)
%p A202385       fi
%p A202385     end:
%p A202385 a:= n-> b(n, n, 0, {}):
%p A202385 seq(a(n), n=31..100);
%t A202385 w[m_, h_] := Product[If[j >= h, 1, j], {j, FactorInteger[m][[All, 1]]}]; b[n_, i_, g_, s_] := b[n, i, g, s] = Module[{j, ok}, Which[n<0, 0, n==0, If[g>1, 0, 1], i<2, 0, True, ok = i <= n; For[j = 1, ok && j <= Length[s], j++, ok = GCD[i, s[[j]]]>1]; b[n, i-1, g, Map[w[#, i]&, s]] + If[ok, b[n-i, i-1, GCD[i, g], Map[w[#, i]&, Union @ Append[s, i]]], 0]]]; a[n_] := b[n, n, 0, {}]; Table[a[n], {n, 31, 100}] (* _Jean-François Alcover_, Feb 15 2017, translated from Maple *)
%Y A202385 Cf. A200976, A018783.
%K A202385 nonn
%O A202385 31,11
%A A202385 _Alois P. Heinz_, Dec 18 2011