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A384033 a(n) is the number of solutions to n = sopfr(k*sopfr(n)) where sopfr(m) is sum of prime factors of m counted with multiplicity.

%I A384033 #55 Jul 21 2025 19:10:30
%S A384033 0,1,1,1,1,0,1,1,1,1,1,2,1,3,4,5,1,7,1,10,10,4,1,19,19,19,30,17,1,40,
%T A384033 1,52,46,12,77,87,1,77,111,87,1,175,1,197,157,197,1,219,302,413,372,
%U A384033 175,1,413,614,413,456,67,1,1083,1,677,819,1552,1552,1695,1
%N A384033 a(n) is the number of solutions to n = sopfr(k*sopfr(n)) where sopfr(m) is sum of prime factors of m counted with multiplicity.
%C A384033 Also, a(n) is the number of star graph labelings with label n on the leaves of the star such that the label of every vertex equals the sum of prime factors (with repetition) over adjacent vertices and all labels are positive.
%H A384033 Robert G. Wilson v, <a href="/A384033/b384033.txt">Table of n, a(n) for n = 1..10000</a>
%H A384033 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a384/A384033.java">Java program</a> (github)
%H A384033 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StarGraph.html">Star graph</a>
%H A384033 Wikipedia, <a href="https://en.wikipedia.org/wiki/Star_(graph_theory)">Star (graph theory)</a>
%H A384033 Robert G. Wilson v, <a href="/A384033/a384033.txt">n, a(n) & center of the star graph</a>
%F A384033 a(n) = A000607(n - sopfr(sopfr(n))) if sopfr(sopfr(n)) <= n, otherwise a(n)=0. - _Sean A. Irvine_, Jun 27 2025
%F A384033 a(p) = 1 if p is a prime. Also, a(n) = 1 when n = {4, 8, 9, 10}. - _Robert G. Wilson v_, Jul 16 2025
%e A384033 A star graph can be defined differently; here, the star graph S_k has a total of k vertices and k-1 leaves.
%e A384033 For all primes n=p as leaf label, the star graph S_2 with 1 central vertex labeled p and 1 leaf labeled p is a solution and the only solution; thus, a(p)=1.
%e A384033   p - p
%e A384033 n=4 as leaf label, works as well with S_2 (2+2=2*2); thus, a(4)=1.
%e A384033   4 - 4
%e A384033 For n=6 as leaf label, there is no respective star graph (as the sum of prime factors is 2+3=5).
%e A384033 n=8 as leaf label can be done as S_4 with the center labeled 18 and the leaves labeled 8 (the prime factor sum for 18=2*3^2 is 2+3+3=8; the prime factor sum for 8=2^3 is 6, adding over all 3 leaves, results in 18); thus, a(8)=1.
%e A384033        8
%e A384033        |
%e A384033   8 -  18 - 8
%e A384033 n=12 is the first labeling with 2 solutions: it can be done with central vertex labeled 35 (S_6) and 42 (S_7); thus, a(12)=2.
%e A384033   The S_6 solution looks like
%e A384033        12   12
%e A384033         \  /
%e A384033     12 - 35 - 12
%e A384033           |
%e A384033          12
%e A384033   as 12=2*2*3 -> 7, 5 leaves sum up to 35 = 5*7 -> 12.
%t A384033 sopfr[n_] := sopfr[n] = Plus @@ Times @@@ FactorInteger@ n; sopfr[1] = 0; t = Series[1/Product[1 - x^Prime[i], {i, 100}], {x, 0, 100}]; a[n_] := Block[{s = sopfr@sopfr@n}, If[s <= n, Coefficient[t, x, n - s], 0]]; Array[a, 67] (* - _Robert G. Wilson v_, Jul 16 2025 *)
%Y A384033 Cf. A000607, A001414.
%K A384033 nonn
%O A384033 1,12
%A A384033 _Michael Terhoeven_, May 17 2025
%E A384033 More terms from _Sean A. Irvine_, Jun 27 2025