A066109 Numbers k such that sigma_4(k)/sigma_2(k) is prime.

4, 9, 20, 25, 169, 289, 961, 1849, 3721, 6889, 11881, 14641, 15625, 17161, 52441, 57121, 66049, 69169, 72361, 96721, 97969, 117649, 130321, 196249, 214369, 253009, 326041, 351649, 358801, 383161, 410881, 418609, 426409, 434281, 491401

Offset: 1

Author

Labos Elemer, Dec 05 2001

Keywords

• nonn

Comments

Numbers k such that A001159(k)/A001157(k) is prime.

Except for the 3rd term 20, below 10000000 all the other terms are even powers of a prime. These primes are listed in A066111. It is not known whether other numbers similar to 20 exist or not.

20 is the only exception within the first 2000 terms. - Amiram Eldar, Feb 25 2025

Examples

			For k = 20: divisors(20) = {20, 10, 5, 4, 2, 1}, sigma_4 = 160000 + 10000 + 625 + 256 + 16 + 1 = 170898, sigma_2 = 400 + 100 + 25 + 16 + 4 + 1 = 546; p = 170898/546 = 73 is prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..2000 (terms 1..250 from Harry J. Smith)

Crossrefs

Cf. A000040, A001157, A001159, A046871, A066111, A066112.

Programs

Mathematica

Do[s = DivisorSigma[4, n]; z = DivisorSigma[2, n]; If[PrimeQ[s/z], Print[{n, s, z, s/z}]], {n, 1, 10000000}]
Select[Range[500000],PrimeQ[DivisorSigma[4,#]/DivisorSigma[2,#]]&] (* Harvey P. Dale, May 02 2011 *)

PARI

isok(k) = { my(f=sigma(k, 4)/sigma(k, 2)); !frac(f) && isprime(f) } \\ Harry J. Smith, Nov 16 2009