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A339312 Sum over all partitions of n of the GCD of the number of parts and the number of distinct parts.

%I A339312 #30 Mar 09 2021 05:07:19
%S A339312 0,1,2,4,6,10,17,23,33,47,71,92,129,169,235,299,408,525,691,885,1147,
%T A339312 1427,1832,2312,2878,3635,4519,5631,7002,8637,10514,13055,15864,19396,
%U A339312 23530,28702,34746,42210,50671,61224,73506,88394,105447,126398,150588,179075
%N A339312 Sum over all partitions of n of the GCD of the number of parts and the number of distinct parts.
%H A339312 Alois P. Heinz, <a href="/A339312/b339312.txt">Table of n, a(n) for n = 0..1000</a>
%p A339312 b:= proc(n, i, p, d) option remember; `if`(n=0, igcd(p, d),
%p A339312       add(b(n-i*j, i-1, p+j, d+signum(j)), j=`if`(i>1, 0..n/i, n)))
%p A339312     end:
%p A339312 a:= n-> b(n$2, 0$2):
%p A339312 seq(a(n), n=0..50);
%t A339312 b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, GCD[p, d],
%t A339312      Sum[b[n - i*j, i - 1, p + j, d + Sign[j]],
%t A339312      {j, If[i > 1, Range[0, n/i], {n}]}]];
%t A339312 a[n_] := b[n, n, 0, 0];
%t A339312 a /@ Range[0, 50] (* _Jean-François Alcover_, Mar 09 2021, after _Alois P. Heinz_ *)
%Y A339312 Cf. A003989, A050873, A096541, A339006, A339011, A339394.
%K A339312 nonn
%O A339312 0,3
%A A339312 _Alois P. Heinz_, Dec 02 2020