A014567 Numbers k such that k and sigma(k) are relatively prime, where sigma(k) = sum of divisors of k (A000203).

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 36, 37, 39, 41, 43, 47, 49, 50, 53, 55, 57, 59, 61, 63, 64, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 89, 93, 97, 98, 100, 101, 103, 107, 109, 111, 113, 115, 119, 121, 125, 127, 128, 129, 131, 133

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy
• nice

Comments

Related to "solitary numbers": n is solitary if there is no other integer m such that sigma(m)/m = sigma(n)/n.

It is easy to show that if n and sigma(n) are relatively prime then n is solitary. But the converse is not true; for example, 18, 45, 48 and 52 are solitary. Probably also 10, 14, 15, 20, 22 and many others are solitary, but I do not think that will ever be proved. - Dean Hickerson

From Daniel Forgues, Jun 23 2009: (Start)

Union of unit, primes and Duffinian numbers.

Duffinian numbers (A003624) are the composite numbers (including, among others, the proper prime powers) for which (n, sigma(n)) = 1. (End)

A009194(a(n)) = 1. - Reinhard Zumkeller, Mar 23 2013

These numbers satisfy (denominator of sigma(n)/n) = n. - Michel Marcus, Oct 27 2013

The asymptotic density of this sequence is 0 (Dressler, 1974; Luca, 2007). - Amiram Eldar, Jul 23 2020

If m*n is in this sequence and gcd(m,n) = 1, then m and n are both in this sequence. - Jianing Song, Aug 07 2022

Examples

			sigma(21) = 1 + 3 + 7 + 21 = 32 is relatively prime to 21, so 21 is in the sequence.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• C. W. Anderson and D. Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84, 65-66, 1977.
• Robert E. Dressler, On a theorem of Niven, Canadian Mathematical Bulletin, Vol. 17, No. 1 (1974), pp. 109-110.
• Andrew Feist, Fun with the sigma(n) function, Missouri Journal of Mathematical Sciences 15:3 (2003), pp. 173-177.
• P. A. Loomis, New families of solitary numbers, J. Algebra and Applications, 14 (No. 9, 2015), #1540004 (6 pages).
• Florian Luca, On the densities of some subsets of integers, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.
• Eric Weisstein's World of Mathematics, Solitary Number.

Crossrefs

Cf. A003624.

Cf. A069059 (complement).

Includes A000961 as a subsequence.

Programs

Haskell

a014567 n = a014567_list !! (n-1)
a014567_list = filter ((== 1) . a009194) [1..]
-- Reinhard Zumkeller, Mar 23 2013

Mathematica

lst={};Do[d=DivisorSigma[1, n];If[GCD[d, n]==1, AppendTo[lst, n]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
Select[Range[150],CoprimeQ[#,DivisorSigma[1,#]]&] (* Harvey P. Dale, Jan 23 2015 *)

PARI

is(n)=gcd(n,sigma(n))==1 \\ Charles R Greathouse IV, Feb 13 2013

Python

from math import gcd
from sympy import divisor_sigma
def ok(n): d = divisor_sigma(n, 1); return gcd(n, d) == 1
print([k for k in range(1, 134) if ok(k)]) # Michael S. Branicky, Mar 28 2022

Formula

a(n) << n log n. Can this be improved? - Charles R Greathouse IV, Feb 13 2013

a(n) >> n log log log n, see Luca. - Charles R Greathouse IV, Feb 17 2014

Extensions

More terms from Labos Elemer