OFFSET
1,12
COMMENTS
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.
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Sean A. Irvine, Java program (github)
Eric Weisstein's World of Mathematics, Star graph
Wikipedia, Star (graph theory)
Robert G. Wilson v, n, a(n) & center of the star graph
FORMULA
a(n) = A000607(n - sopfr(sopfr(n))) if sopfr(sopfr(n)) <= n, otherwise a(n)=0. - Sean A. Irvine, Jun 27 2025
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
EXAMPLE
A star graph can be defined differently; here, the star graph S_k has a total of k vertices and k-1 leaves.
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.
p - p
n=4 as leaf label, works as well with S_2 (2+2=2*2); thus, a(4)=1.
4 - 4
For n=6 as leaf label, there is no respective star graph (as the sum of prime factors is 2+3=5).
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.
8
|
8 - 18 - 8
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.
The S_6 solution looks like
12 12
\ /
12 - 35 - 12
|
12
as 12=2*2*3 -> 7, 5 leaves sum up to 35 = 5*7 -> 12.
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Terhoeven, May 17 2025
EXTENSIONS
More terms from Sean A. Irvine, Jun 27 2025
STATUS
approved