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 A258913 #32 Dec 16 2024 01:53:52
%S A258913 1,0,2,3,0,5,4,7,0,0,0,17,9,13,8,0,0,27,0,19,0,0,0,52,0,0,0,12,0,29,
%T A258913 41,52,0,0,0,22,0,37,18,27,0,87,0,43,0,0,0,115,0,0,0,0,0,87,0,67,49,0,
%U A258913 0,121,0,61,32,0,0,0,0,67,0,0,0,253,0,73,0,0,0
%N A258913 a(n) is the sum of all numbers k for which sigma(k) = n.
%C A258913 Here sigma is A000203, the sum-of-divisors function.
%C A258913 a(n) is the sum of the n-th row in A085790.
%C A258913 We can divide the set of natural numbers into three classes based on whether a(n)<n, a(n)=n, or a(n)>n. The last class is A258914. Are there any n in the second category, i.e., n such that a(n)=n, other than n=1 (see link)?
%C A258913 It is natural to further divide the class a(n)<n into two subclasses: a(n)=0 and 0<a(n)<n. The first one corresponds exactly to A007369 (not in image of sigma), which is all n for which A054973(n)=0. The second one of these, the case 0<a(n)<n, includes (for n>1) all of A007370 (just one pre-image of n under sigma, equivalently A054973(n)=1), but also includes some terms that have more than one pre-image, see A258931.
%C A258913 If there exists a number n>1 such that a(n)=n, then n > 2.5*10^10. - _Giovanni Resta_, Jun 15 2015
%C A258913 Row sums of A299762. - _Omar E. Pol_, Mar 14 2018
%H A258913 Charles R Greathouse IV, <a href="/A258913/b258913.txt">Table of n, a(n) for n = 1..10000</a>
%H A258913 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A258913 Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/1126988/">A number n which is the sum of all numbers k with sigma(k)=n?</a>
%e A258913 To find a(24), note that the only values of k with sigma(k)=24 are k=14,15,23; therefore a(24)=14+15+23=52.
%t A258913 a[n_] := Sum[k*Boole[DivisorSigma[1, k] == n], {k, 1, n}]; Array[a, 80] (* _Jean-François Alcover_, Jun 15 2015 *)
%o A258913 (PARI) a(n)=sum(k=1,n,if(sigma(k)==n,k))
%o A258913 (PARI) first(n)=my(v=vector(n),s); for(k=1,n,s=sigma(k);if(s<=n,v[s]+=k));v \\ _Charles R Greathouse IV_, Jun 15 2015
%o A258913 (PARI) a(n) = vecsum(invsigma(n)); \\ _Amiram Eldar_, Dec 16 2024, using _Max Alekseyev_'s invphi.gp
%Y A258913 Cf. A000203, A007369, A007370, A054973, A085790, A258914, A258931, A299762.
%K A258913 nonn
%O A258913 1,3
%A A258913 _Jeppe Stig Nielsen_, Jun 14 2015