

A014567


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


50



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
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OFFSET

1,2


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


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, 6566, 1977.
Robert E. Dressler, On a theorem of Niven, Canadian Mathematical Bulletin, Vol. 17, No. 1 (1974), pp. 109110.
Andrew Feist, Fun with the sigma(n) function, Missouri Journal of Mathematical Sciences 15:3 (2003), pp. 173177.
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. 167170.
Eric Weisstein's World of Mathematics, Solitary Number.


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


EXAMPLE

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


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 *)


PROG

(PARI) is(n)=gcd(n, sigma(n))==1 \\ Charles R Greathouse IV, Feb 13 2013
(Haskell)
a014567 n = a014567_list !! (n1)
a014567_list = filter ((== 1) . a009194) [1..]
 Reinhard Zumkeller, Mar 23 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


CROSSREFS

Cf. A003624.
Cf. A069059 (complement).
Includes A000961 as a subsequence.
Sequence in context: A344979 A273130 A273200 * A324769 A328867 A326536
Adjacent sequences: A014564 A014565 A014566 * A014568 A014569 A014570


KEYWORD

nonn,easy,nice


AUTHOR

Eric W. Weisstein


EXTENSIONS

More terms from Labos Elemer


STATUS

approved



