A053222 - OEIS

%I #52 Oct 27 2023 22:00:45

%S 2,1,3,-1,6,-4,7,-2,5,-6,16,-14,10,0,7,-13,21,-19,22,-10,4,-12,36,-29,

%T 11,-2,16,-26,42,-40,31,-15,6,-6,43,-53,22,-4,34,-48,54,-52,40,-6,-6,

%U -24,76,-67,36,-21,26,-44,66,-48,48,-40,10,-30,108,-106,34,8,23,-43,60,-76,58

%N First differences of sigma(n).

%C a(A002961(n)) = 0. - _Reinhard Zumkeller_, Dec 28 2011

%C 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

%H T. D. Noe, <a href="/A053222/b053222.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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

%F G.f.: -1 + (1 - x)*Sum_{k>=1} k*x^(k-1)/(1 - x^k). - _Ilya Gutkovskiy_, Jan 29 2017

%p A053222 := proc(n)

%p numtheory[sigma](n+1)-numtheory[sigma](n) ;

%p end proc: # _R. J. Mathar_, Jul 08 2013

%t DivisorSigma[1, Range[100]] // Differences (* _Jean-François Alcover_, Jan 26 2018 *)

%o (Haskell)

%o a053222 n = a053222_list !! (n-1)

%o a053222_list = zipWith (-) (tail a000203_list) a000203_list

%o -- _Reinhard Zumkeller_, Oct 16 2011

%o (PARI) a(n)=sigma(n+1)-sigma(n) \\ _Charles R Greathouse IV_, Mar 09 2014

%o (GAP) List([1..70], n -> Sigma(n+1)-Sigma(n)); # _Muniru A Asiru_, Feb 14 2018

%o (Magma) [DivisorSigma(1, n+1) - DivisorSigma(1,n): n in [1..100]]; // _G. C. Greubel_, Sep 03 2018

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

%K sign,look

%O 1,1

%A _Asher Auel_, Jan 06 2000