A219545 Integer values of sigma(n)/n that are prime.

2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 5, 5, 3, 2, 5, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 2, 5, 7, 2, 5, 5, 7, 5, 5, 5, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 5, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Author

Jonathan Sondow, Nov 22 2012

Keywords

• nonn

Comments

Subsequence of A054030 consisting of primes among the abundancies sigma(m)/m of multiply perfect numbers m (see A007691).

Each 2 corresponds to a perfect number A000396, so if there are infinitely many perfect numbers, then the sequence is infinite.

If, in addition, there are only finitely many multiply perfect numbers m with sigma(m)/m > 2 (see A134639), then a(n) = 2 for all n > some N.

Examples

			A065997(1) = 6 and sigma(6)/6 = (1+2+3+6)/6 = 2, so a(1) = 2.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, B2.

Links

• A. Flammenkamp, The Multiply Perfect Numbers Page

Crossrefs

Cf. A000396, A007691, A054030, A065997, A134639.

Programs

Mathematica

Select[Table[DivisorSigma[1,n]/n,{n,10^6}],PrimeQ] (* The program only generates the first seven terms of the sequence. To generate them all, the value of n would have to be greatly increased. *) (* Harvey P. Dale, Oct 25 2021 *)

Formula

a(n) = sigma(A065997(n))/A065997(n).

Extensions

Extended by T. D. Noe, Nov 27 2012