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.

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

Antti Karttunen, Table of n, a(n) for n = 1..16384

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: A125250 A048113 A028961 * A036273 A342563 A343633

Adjacent sequences: A110174 A110175 A110176 * A110178 A110179 A110180

Keyword

nonn

Author

T. D. Noe, Jul 15 2005

Status

approved