A095738 Numbers that are coprime to sigma but are not prime powers.

21, 35, 36, 39, 50, 55, 57, 63, 65, 75, 77, 85, 93, 98, 100, 111, 115, 119, 129, 133, 143, 144, 155, 161, 171, 175, 183, 185, 187, 189, 201, 203, 205, 209, 215, 217, 219, 221, 225, 235, 237, 242, 245, 247, 253, 259, 265, 275, 279, 291, 299, 301, 305, 309, 319

Offset: 1

Author

Walter Nissen, Jul 08 2004

Keywords

• nonn

Comments

Abundancy is defined as the ratio of the multiplicative sum-of-divisors function to the integer itself: abund(n) = sigma(n)/n. E.g., abund(10) = sigma(10) / 10 = (1+2+5+10) / 10 = 1.8 = 9 / 5.

Integers m and n are friendly if and only if they have the same abundancy. E.g., abund(12) = abund(234) = 7 / 3, so 12 and 234 are friends.

Integers which have no friends are called solitary.

The numbers in this sequence are solitary.

Compare abundancy to abundance as defined in A033880.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Claude W. Anderson and Dean Hickerson, Advanced Problem 6020: Friendly Integers, Amer. Math. Monthly, 1977, V84#1p65-6.
• Walter Nissen, Primitive Friendly Integers and Exclusive Multiples, 2004 post to NMBRTHRY mailing list

Crossrefs

Cf. A014567, A074902, A095739, A000203, A033880.

Programs

Mathematica

Select[Range[320], PrimeNu[#] > 1 && GCD[#, DivisorSigma[1, #]] == 1 &] (* Amiram Eldar, Jun 25 2019 *)

PARI

isok(n) = (gcd(sigma(n), n) == 1) && (! isprime(n)) && (! (ispower(n, , &p) && isprime(p))); \\ Michel Marcus, Jan 24 2014

Extensions

Edited by Franklin T. Adams-Watters, Mar 06 2014