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A053222 First differences of sigma(n).

%I A053222 #52 Oct 27 2023 22:00:45
%S A053222 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 A053222 11,-2,16,-26,42,-40,31,-15,6,-6,43,-53,22,-4,34,-48,54,-52,40,-6,-6,
%U A053222 -24,76,-67,36,-21,26,-44,66,-48,48,-40,10,-30,108,-106,34,8,23,-43,60,-76,58
%N A053222 First differences of sigma(n).
%C A053222 a(A002961(n)) = 0. - _Reinhard Zumkeller_, Dec 28 2011
%C A053222 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 A053222 T. D. Noe, <a href="/A053222/b053222.txt">Table of n, a(n) for n = 1..10000</a>
%F A053222 a(n) = A000203(n+1) - A000203(n).
%F A053222 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 A053222 G.f.: -1 + (1 - x)*Sum_{k>=1} k*x^(k-1)/(1 - x^k). - _Ilya Gutkovskiy_, Jan 29 2017
%p A053222 A053222 := proc(n)
%p A053222 numtheory[sigma](n+1)-numtheory[sigma](n) ;
%p A053222 end proc: # _R. J. Mathar_, Jul 08 2013
%t A053222 DivisorSigma[1, Range[100]] // Differences (* _Jean-François Alcover_, Jan 26 2018 *)
%o A053222 (Haskell)
%o A053222 a053222 n = a053222_list !! (n-1)
%o A053222 a053222_list = zipWith (-) (tail a000203_list) a000203_list
%o A053222 -- _Reinhard Zumkeller_, Oct 16 2011
%o A053222 (PARI) a(n)=sigma(n+1)-sigma(n) \\ _Charles R Greathouse IV_, Mar 09 2014
%o A053222 (GAP) List([1..70], n -> Sigma(n+1)-Sigma(n)); # _Muniru A Asiru_, Feb 14 2018
%o A053222 (Magma) [DivisorSigma(1, n+1) - DivisorSigma(1,n): n in [1..100]]; // _G. C. Greubel_, Sep 03 2018
%Y A053222 Cf. A000203, A053223, A053225, A053227, A053224, A053226.
%K A053222 sign,look
%O A053222 1,1
%A A053222 _Asher Auel_, Jan 06 2000