This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A155085 #62 Jul 09 2025 04:30:34
%S A155085 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 A155085 56,68,67,84,59,102,63,95,81,88,83,127,75,98,95,130,83,138,87,128,123,
%U A155085 118,95,172,106,143,123,150,107,174,127,176,137,148,119,228,123,158,167,191
%N A155085 a(n) = n + sum of divisors of n.
%C A155085 a(n) is a perfect number for n = 5 and n = r*q with r = 4*46817 and q = 4477417228433 = (A006516(31)-sigma(r))/a(r) [Radcliffe, 2025]. Are there other n with this property? - _M. F. Hasler_, Mar 10 2025
%H A155085 Antti Karttunen, <a href="/A155085/b155085.txt">Table of n, a(n) for n = 1..20000</a>
%H A155085 David Radcliffe, in reply to Leo Hennig, <a href="https://groups.google.com/g/seqfan/c/AWt75irA3TY/m/Smz-0ON9AgAJ">Re: One entry and just one entry</a>, SeqFan list, March 6, 2025.
%F A155085 If n is a prime, a(n) = 2n+1.
%F A155085 If n is a perfect number, a(n) = 3n.
%F A155085 If n is of the form 2^m, a(n) = 2^(m+1) + 2^m -1.
%F A155085 Generally a(n) >= 2n+1.
%F A155085 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 A155085 a(n) = A000203(n) + n = A001065(n) + 2*n. - _Michael Somos_, Sep 19 2011
%F A155085 a(n) = A001065(-n). - _Michael Somos_, Sep 20 2011
%F A155085 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 A155085 G.f.: x/(1 - x)^2 + Sum_{k>=1} x^k/(1 - x^k)^2. - _Ilya Gutkovskiy_, Mar 17 2017
%F A155085 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 A155085 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.
%t A155085 Table[n+DivisorSigma[1,n],{n,70}] (* _Harvey P. Dale_, Apr 27 2019 *)
%o A155085 (PARI) a(n) = sigma(n) + n; /* _Michael Somos_, Sep 19 2011 */
%Y A155085 Cf. A000203, A001065, A013661.
%K A155085 nonn
%O A155085 1,1
%A A155085 Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009
%E A155085 More terms from _N. J. A. Sloane_, Jan 24 2009
%E A155085 Zero removed and offset corrected by _Omar E. Pol_, Jan 27 2009