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A309915 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of n and m runs through the set of least numbers whose prime signature is a partition of 2n.

%I A309915 #15 Dec 12 2020 14:38:35
%S A309915 1,3,16,79,371,1683,7413,31769,133692,553848,2265776,9181670,36928673,
%T A309915 147650125,587734595,2331625130,9226486717,36443758767,143763811785,
%U A309915 566624864014,2232055573265,8789903797692,34610963678036,136287108614677,536724439657635
%N A309915 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of n and m runs through the set of least numbers whose prime signature is a partition of 2n.
%H A309915 Alois P. Heinz, <a href="/A309915/b309915.txt">Table of n, a(n) for n = 0..380</a>
%F A309915 a(n) = A079025(2n,n).
%e A309915 a(2) = 16: The partitions of 2*2 are (4), (31), (22), (211), (1111). Least numbers with these prime signatures are 16, 24, 36, 60, 210. Their divisors with prime signatures (2) or (11) are {4}, {4,6}, {4,6,9}, {4,6,10,15}, {6,10,14,15,21,35}. The total number is 1 + 2 + 3 + 4 + 6.
%p A309915 b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (x+1)^n,
%p A309915       b(n, i-1) +factor((x^(i+1)-1)/(x-1))*b(n-i, min(n-i, i))))
%p A309915     end:
%p A309915 a:= n-> coeff(b(2*n$2), x, n):
%p A309915 seq(a(n), n=0..25);
%t A309915 b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + Factor[(x^(i + 1) - 1)/(x - 1)] b[n - i, Min[n - i, i]]]];
%t A309915 a[n_] := Coefficient[b[2n, 2n], x, n];
%t A309915 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 12 2020, after _Alois P. Heinz_ *)
%Y A309915 Cf. A000041, A079025, A087443.
%K A309915 nonn
%O A309915 0,2
%A A309915 _Alois P. Heinz_, Aug 22 2019