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

1,1

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A144205 A261408 A008946 * A216467 A330949 A248020

Adjacent sequences: A095735 A095736 A095737 * A095739 A095740 A095741

Keyword

nonn

Author

Walter Nissen, Jul 08 2004

Extensions

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

Status

approved