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A385646 a(n) is the number of distinct sums of distinct prime factors of n.

%I A385646 #5 Jul 15 2025 16:46:04
%S A385646 0,1,1,1,1,3,1,1,1,3,1,3,1,3,3,1,1,3,1,3,3,3,1,3,1,3,1,3,1,6,1,1,3,3,
%T A385646 3,3,1,3,3,3,1,7,1,3,3,3,1,3,1,3,3,3,1,3,3,3,3,3,1,6,1,3,3,1,3,7,1,3,
%U A385646 3,6,1,3,1,3,3,3,3,7,1,3,1,3,1,7,3,3,3,3
%N A385646 a(n) is the number of distinct sums of distinct prime factors of n.
%H A385646 Felix Huber, <a href="/A385646/b385646.txt">Table of n, a(n) for n = 1..10000</a>
%F A385646 a(n) < A385646(n).
%e A385646 The a(18) = 3 distinct sums of distinct prime factors of 18 = 2*3^2 are 2, 3 and 2 + 3.
%e A385646 The a(42) = 7 distinct sums of distinct prime factors of 42 = 2*3*7 are 2, 3, 7, 2 + 3 = 5, 2 + 7 = 9, 3 + 7 = 10, 2 + 3 + 7 = 12.
%e A385646 The a(30) = 6 distinct sums of distinct prime factors of 30 = 2*3*5 are 2, 3, 2 + 3 = 5, 2 + 5 = 7, 3 + 5 = 8, 2 + 3 + 5 = 10.
%p A385646 A385646:=proc(n)
%p A385646     local b,k,l,i,j;
%p A385646     l:=[seq(i[1],i in ifactors(n)[2])]:
%p A385646     b:=proc(m,i)
%p A385646         option remember;
%p A385646         `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
%p A385646     end;
%p A385646     return nops(select(x->x>0,[seq(b(k,nops(l)),k=1..add(l))]))
%p A385646     end:
%p A385646 seq(A385646(n),n=1..85);
%Y A385646 Cf. A000040, A001221, A119347, A385645.
%K A385646 nonn,easy
%O A385646 1,6
%A A385646 _Felix Huber_, Jul 11 2025