OFFSET
0,3
COMMENTS
Call an integer k between 1 and n a "semi-divisor" of n if n leaves a remainder of 1 when divided by k, i.e., n == 1 (mod k). a(n) gives the sum of the semi-divisors of n+1. - Joseph L. Pe, Sep 11 2002
a(n) is also the sum of the strong divisors of n, for n >= 1. - Omar E. Pol, May 01 2015
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..10000
FORMULA
a(p) = p for p prime.
G.f.: -2*x^2/(Q(0) - 2*x^2 + 2*x), where Q(k) = (2*x^(k+2) - x - 1)*k - 1 - 2*x + 3*x^(k+2) - x*(k+3)*(k+1)*(1-x^(k+2))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 16 2013
Let A(x) be the g.f. of A039653 and B(x) the g.f. of A155085. Then B(x) = 1/(1-x) + 1/(1-x)^2 + A(x)/x. - Sergei N. Gladkovskii, May 16 2013
MAPLE
with(numtheory): A039653:=n->sigma(n)-1: (0, seq(A039653(n), n=1..100)); # Wesley Ivan Hurt, Jul 09 2015
MATHEMATICA
Join[{0}, Table[DivisorSigma[1, n] - 1, {n, 90}]] (* Vincenzo Librandi, May 02 2015 *)
PROG
(Magma) [0] cat [DivisorSigma(1, n)-1: n in [1..100]]; // Vincenzo Librandi, May 02 2015
(PARI) A039653(n) = if(!n, n, sigma(n)-1); \\ Antti Karttunen, May 26 2017
(Python)
from sympy import divisor_sigma
def A039653(n): return divisor_sigma(n)-1 if n else 0 # Chai Wah Wu, Mar 14 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved