A193066 -id:A193066 - cp's OEIS frontend

A193065 Odd numbers N for which numerator(sigma(N)/N) is a prime.

9, 25, 289, 729, 1521, 1681, 2401, 3481, 5041, 7921, 10201, 15625, 17161, 27889, 28561, 29929, 83521, 85849, 146689, 257049, 279841, 458329, 491401, 531441, 552049, 579121, 597529, 683929, 703921, 707281, 734449, 829921, 1190281, 1203409, 1352569, 1394761, 1423249, 1481089, 1885129, 2036329, 2211169

Offset: 1

M. F. Hasler, Jul 15 2011

nonn

This sequence includes all odd terms of A023194.
For most of the terms, sigma(N) is prime (i.e., N is in A023194); the first two exceptions are sigma(a(5))=3*13*61 and sigma(a(20))=13*30941. See A193072 for (the square root of) these exceptions.
It is well known that sigma(N) can't be odd unless N is a square (since sigma is multiplicative and sigma(p^e)=1+...+p^e) or twice a square (excluded here).
See A193066 for the square roots of the terms.
The sequence of numbers n for which A002129(n) is prime starts as this sequence here, but excludes a(5), a(20) etc. - R. J. Mathar, Sep 18 2011

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000203.

Select[Range[1,23*10^5,2],PrimeQ[Numerator[DivisorSigma[1,#]/#]]&] (* Harvey P. Dale, Sep 17 2017 *)

forstep(N=1,1e7,2,isprime(numerator(sigma(N)/N)) && print1(N","))

a(n) = A193066(n)^2.

A193072 Odd numbers N for which numerator(sigma(N^2)/N^2) is prime but sigma(N^2) is composite.

Original entry on oeis.org

39, 507, 2379, 6591, 13167, 29511, 148955, 1672209, 8852259, 212370543, 1929229929

Offset: 1

M. F. Hasler, Jul 15 2011

a(12) > 10^10. - Lucas A. Brown, Apr 12 2021

Lucas A. Brown, A193072.py

Intersection of A193066 and A193071.

forstep(N=1, 1e7, 2, isprime(numerator(sigma(N^2)/N^2)) && !isprime(sigma(N^2)) && print1(N", "))

a(8)-a(10) from Donovan Johnson, Sep 19 2011

a(11) from Lucas A. Brown, Apr 12 2021

cp's OEIS Frontend

A193065 Odd numbers N for which numerator(sigma(N)/N) is a prime.

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