A039653 a(0) = 0; for n > 0, a(n) = sigma(n)-1.

0, 0, 2, 3, 6, 5, 11, 7, 14, 12, 17, 11, 27, 13, 23, 23, 30, 17, 38, 19, 41, 31, 35, 23, 59, 30, 41, 39, 55, 29, 71, 31, 62, 47, 53, 47, 90, 37, 59, 55, 89, 41, 95, 43, 83, 77, 71, 47, 123, 56, 92, 71, 97, 53, 119, 71, 119, 79, 89, 59, 167, 61, 95, 103, 126, 83, 143, 67, 125, 95

Offset: 0

Author

David W. Wilson

Keywords

• nonn

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
• OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Crossrefs

Cf. A000203, A039649, A039650, A039651, A039652, A039653, A039654, A039655, A039656, A032741.

Programs

Magma

[0] cat [DivisorSigma(1, n)-1: n in [1..100]]; // Vincenzo Librandi, May 02 2015

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 *)

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

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