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A202523 Number of partitions of n into distinct parts having pairwise prime GCDs but no overall common factor.

%I A202523 #21 Dec 05 2020 14:46:13
%S A202523 1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,0,1,0,0,0,2,0,1,0,0,0,2,0,4,0,0,0,
%T A202523 1,0,4,0,1,0,5,0,4,0,0,0,3,0,5,0,0,0,6,0,1,1,2,0,6,0,6,1,1,0,4,0,12,0,
%U A202523 1,1,12,1,9,0,0,1,10,0,10,0,1,2,10,1,4
%N A202523 Number of partitions of n into distinct parts having pairwise prime GCDs but no overall common factor.
%H A202523 Alois P. Heinz, <a href="/A202523/b202523.txt">Table of n, a(n) for n = 31..251</a>
%e A202523 a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
%e A202523 a(37) = 1: [10,12,15] = [2*5,2*2*3,3*5].
%e A202523 a(41) = 2: [6,15,20], [6,14,21].
%e A202523 a(43) = 1: [10,15,18].
%e A202523 a(47) = 1: [12,14,21].
%e A202523 a(49) = 1: [10,15,24].
%e A202523 a(61) = 4: [6,22,33], [10,15,36], [6,15,40], [6,10,45].
%p A202523 w:=(m, h)-> mul(`if`(j[1]>=h, 1, j[1]^min(j[2], 2)), j=ifactors(m)[2]):
%p A202523 b:= proc(n, i, g, s) option remember; local j, ok, si;
%p A202523       if n=0 then `if`(g>1, 0, 1)
%p A202523     elif i<2 or member(1, s) then 0
%p A202523     else ok:= evalb(i<=n);
%p A202523          si:= map(x->w(x, i), s);
%p A202523          for j in s while ok do ok:= isprime(igcd(i, j)) od;
%p A202523          b(n, i-1, g, si) +`if`(ok,
%p A202523          b(n-i, i-1, igcd(i, g), si union {w(i, i)} ), 0)
%p A202523       fi
%p A202523     end:
%p A202523 a:= n-> b(n, n, 0, {}):
%p A202523 seq(a(n), n=31..100);
%t A202523 w[m_, h_] := Product[If[j[[1]] >= h, 1, j[[1]]^Min[j[[2]], 2]], {j, FactorInteger[m]}];
%t A202523 b[n_, i_, g_, s_] := b[n, i, g, s] = Module[{j, ok, si}, Which[n == 0, If[g > 1, 0, 1], i < 2 || MemberQ[s, 1], 0, True, ok = i <= n; si = w[#, i]& /@ s; For[j = 1, j <= Length[s], j++, If[!ok, Break[]]; ok = PrimeQ[ GCD[i, s[[j]]]]]; b[n, i - 1, g, si] + If[ok, b[n - i, i - 1, GCD[i, g], si ~Union~ {w[i, i]}], 0]]];
%t A202523 a[n_] := b[n, n, 0, {}];
%t A202523 Table[a[n], {n, 31, 100}] (* _Jean-François Alcover_, Dec 05 2020, after _Alois P. Heinz_ *)
%Y A202523 Cf. A202385, A202425, A200976, A018783.
%K A202523 nonn
%O A202523 31,11
%A A202523 _Alois P. Heinz_, Dec 20 2011