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%I #46 Mar 17 2024 05:07:41
%S 2,5,7,11,11,18,15,23,22,28,23,40,27,38,39,47,35,57,39,62,53,58,47,84,
%T 56,68,67,84,59,102,63,95,81,88,83,127,75,98,95,130,83,138,87,128,123,
%U 118,95,172,106,143,123,150,107,174,127,176,137,148,119,228,123,158,167,191
%N a(n) = n + sum of divisors of n.
%C If n is a prime, a(n)=2n+1, if n is a perfect number a(n)=3n, if n is of the form 2^m, a(n) = 2^(m+1) + 2^m -1. Generally a(n)>=2n+1.
%H Antti Karttunen, <a href="/A155085/b155085.txt">Table of n, a(n) for n = 1..20000</a>
%F If n = Product_{i=1...k} Pi^ki is the prime power factorization of n, then a(n) = n + [Product_{i=1...k} {Pi^(ki+1)-1}]/[Product_{i=1...k} (Pi-1)]. For example, 18 = 2*3^2; a(18) = 18+(2^2-1)*(3^3-1)/[(2-1)*(3-1)] = 57.
%F a(n) = A000203(n) + n = A001065(n) + 2*n. - _Michael Somos_, Sep 19 2011
%F a(n) = A001065(-n). - _Michael Somos_, Sep 20 2011
%F G.f.: 1/(1-x)+1/(1-x)^2 - 2*x/(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.: x/(1 - x)^2 + Sum_{k>=1} x^k/(1 - x^k)^2. - _Ilya Gutkovskiy_, Mar 17 2017
%F Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = (zeta(2)+1)/2 = 1.322467... . - _Amiram Eldar_, Mar 17 2024
%e Taking 18 as an example, a(18) = 18+1+2+3+6+9+18=57; 18=2*3^2; a(18)=18+(2^2-1)*(3^3-1)/[(2-1)*(3-1)]=57. [Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 25 2009]
%t Table[n+DivisorSigma[1,n],{n,70}] (* _Harvey P. Dale_, Apr 27 2019 *)
%o (PARI) {a(n) = if( n==0, 0, sigma(n) + n)}; /* _Michael Somos_, Sep 19 2011 */
%Y Cf. A000203, A001065, A013661.
%K nonn
%O 1,1
%A Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009, Jan 25 2009, Jan 26 2009
%E More terms from _N. J. A. Sloane_, Jan 24 2009
%E Zero removed and offset corrected by _Omar E. Pol_, Jan 27 2009
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