OFFSET
1,3
COMMENTS
Here sigma is A000203, the sum-of-divisors function.
a(n) is the sum of the n-th row in A085790.
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)?
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.
If there exists a number n>1 such that a(n)=n, then n > 2.5*10^10. - Giovanni Resta, Jun 15 2015
Row sums of A299762. - Omar E. Pol, Mar 14 2018
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Mathematics Stack Exchange, A number n which is the sum of all numbers k with sigma(k)=n?
EXAMPLE
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.
MATHEMATICA
a[n_] := Sum[k*Boole[DivisorSigma[1, k] == n], {k, 1, n}]; Array[a, 80] (* Jean-François Alcover, Jun 15 2015 *)
PROG
(PARI) a(n)=sum(k=1, n, if(sigma(k)==n, k))
(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
(PARI) a(n) = vecsum(invsigma(n)); \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Jeppe Stig Nielsen, Jun 14 2015
STATUS
approved