A053222 First differences of sigma(n).

2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8, 23, -43, 60, -76, 58

Offset: 1

Author

Asher Auel, Jan 06 2000

Keywords

• sign
• look

Comments

a(A002961(n)) = 0. - Reinhard Zumkeller, Dec 28 2011

Considering the values |a(n)| <= 100 for n < 10^13, we notice that some odd values do not appear within that range, namely 9, 17, 25, 27, 33, 37, 39, 45, 47, 49, 51, 55, 57, 59, 69, 71, 77, 81, 83, 87, 89, 91, 95, 97, and 99. All the other absolute values <= 100 appear for n < 3600, with the exception of a(1159742043) = 62. - Giovanni Resta, Jun 26 2017

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A053223, A053225, A053227, A053224, A053226.

Programs

GAP

List([1..70], n -> Sigma(n+1)-Sigma(n)); # Muniru A Asiru, Feb 14 2018

Haskell

a053222 n = a053222_list !! (n-1)
a053222_list = zipWith (-) (tail a000203_list) a000203_list
-- Reinhard Zumkeller, Oct 16 2011

Magma

[DivisorSigma(1, n+1) - DivisorSigma(1,n): n in [1..100]]; // G. C. Greubel, Sep 03 2018

Maple

A053222 := proc(n)
numtheory[sigma](n+1)-numtheory[sigma](n) ;
end proc: # R. J. Mathar, Jul 08 2013

Mathematica

DivisorSigma[1, Range[100]] // Differences (* Jean-François Alcover, Jan 26 2018 *)

PARI

a(n)=sigma(n+1)-sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

Formula

a(n) = A000203(n+1) - A000203(n).

G.f.: 2*(x-1)/(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

G.f.: -1 + (1 - x)*Sum_{k>=1} k*x^(k-1)/(1 - x^k). - Ilya Gutkovskiy, Jan 29 2017