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A110177
Number of solutions 0<k<n to the equation sigma(n) = sigma(k) + sigma(n-k), where sigma is the sum of divisors function.
3
0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 2, 0, 2, 0, 2, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 4
OFFSET
1,3
COMMENTS
The number of solutions is always even because k=n/2 cannot be a solution for even n.
LINKS
MATHEMATICA
a[n_] := Select[Range[n-1], DivisorSigma[1, n]==DivisorSigma[1, n-# ]+DivisorSigma[1, # ]&]; Table[Length[a[n]], {n, 150}]
PROG
(PARI) A110177(n) = { my(x=sigma(n)); sum(k=1, n-1, (x == (sigma(k)+sigma(n-k)))); }; \\ Antti Karttunen, Feb 20 2023
CROSSREFS
Cf. A110176 (least k such that sigma(n)=sigma(k)+sigma(n-k)).
Cf. also A110174.
Sequence in context: A048113 A028961 A373242 * A036273 A342563 A343633
KEYWORD
nonn
AUTHOR
T. D. Noe, Jul 15 2005
STATUS
approved