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

1,1

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

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

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 *)

Prog

(Haskell)
a053222 n = a053222_list !! (n-1)
a053222_list = zipWith (-) (tail a000203_list) a000203_list
-- Reinhard Zumkeller, Oct 16 2011
(PARI) a(n)=sigma(n+1)-sigma(n) \\ Charles R Greathouse IV, Mar 09 2014
(GAP) List([1..70], n -> Sigma(n+1)-Sigma(n)); # Muniru A Asiru, Feb 14 2018
(Magma) [DivisorSigma(1, n+1) - DivisorSigma(1, n): n in [1..100]]; // G. C. Greubel, Sep 03 2018

Crossrefs

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

Sequence in context: A034869 A205858 A340793 * A351159 A318762 A262598

Adjacent sequences: A053219 A053220 A053221 * A053223 A053224 A053225

Keyword

sign,look

Author

Asher Auel, Jan 06 2000

Status

approved