cp's OEIS Frontend

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

%I A014567 #61 Feb 16 2025 08:32:33
%S A014567 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,
%T A014567 43,47,49,50,53,55,57,59,61,63,64,65,67,71,73,75,77,79,81,83,85,89,93,
%U A014567 97,98,100,101,103,107,109,111,113,115,119,121,125,127,128,129,131,133
%N A014567 Numbers k such that k and sigma(k) are relatively prime, where sigma(k) = sum of divisors of k (A000203).
%C A014567 Related to "solitary numbers": n is solitary if there is no other integer m such that sigma(m)/m = sigma(n)/n.
%C A014567 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_
%C A014567 From _Daniel Forgues_, Jun 23 2009: (Start)
%C A014567 Union of unit, primes and Duffinian numbers.
%C A014567 Duffinian numbers (A003624) are the composite numbers (including, among others, the proper prime powers) for which (n, sigma(n)) = 1. (End)
%C A014567 A009194(a(n)) = 1. - _Reinhard Zumkeller_, Mar 23 2013
%C A014567 These numbers satisfy (denominator of sigma(n)/n) = n. - _Michel Marcus_, Oct 27 2013
%C A014567 The asymptotic density of this sequence is 0 (Dressler, 1974; Luca, 2007). - _Amiram Eldar_, Jul 23 2020
%C A014567 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
%H A014567 T. D. Noe, <a href="/A014567/b014567.txt">Table of n, a(n) for n = 1..1000</a>
%H A014567 C. W. Anderson and D. Hickerson, <a href="http://www.jstor.org/stable/2318325">Problem 6020: Friendly Integers</a>, Amer. Math. Monthly 84, 65-66, 1977.
%H A014567 Robert E. Dressler, <a href="https://doi.org/10.4153/CMB-1974-019-5">On a theorem of Niven</a>, Canadian Mathematical Bulletin, Vol. 17, No. 1 (1974), pp. 109-110.
%H A014567 Andrew Feist, <a href="http://www.math-cs.ucmo.edu/~mjms/2003.3/feist.pdf">Fun with the sigma(n) function</a>, Missouri Journal of Mathematical Sciences 15:3 (2003), pp. 173-177.
%H A014567 P. A. Loomis, <a href="http://dx.doi.org/10.1142/S0219498815400046">New families of solitary numbers</a>, J. Algebra and Applications, 14 (No. 9, 2015), #1540004 (6 pages).
%H A014567 Florian Luca, <a href="http://projecteuclid.org/euclid.mjms/1316032973">On the densities of some subsets of integers</a>, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.
%H A014567 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SolitaryNumber.html">Solitary Number.</a>
%F A014567 a(n) << n log n. Can this be improved? - _Charles R Greathouse IV_, Feb 13 2013
%F A014567 a(n) >> n log log log n, see Luca. - _Charles R Greathouse IV_, Feb 17 2014
%e A014567 sigma(21) = 1 + 3 + 7 + 21 = 32 is relatively prime to 21, so 21 is in the sequence.
%t A014567 lst={};Do[d=DivisorSigma[1, n];If[GCD[d, n]==1, AppendTo[lst, n]], {n, 6!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 01 2008 *)
%t A014567 Select[Range[150],CoprimeQ[#,DivisorSigma[1,#]]&] (* _Harvey P. Dale_, Jan 23 2015 *)
%o A014567 (PARI) is(n)=gcd(n,sigma(n))==1 \\ _Charles R Greathouse IV_, Feb 13 2013
%o A014567 (Haskell)
%o A014567 a014567 n = a014567_list !! (n-1)
%o A014567 a014567_list = filter ((== 1) . a009194) [1..]
%o A014567 -- _Reinhard Zumkeller_, Mar 23 2013
%o A014567 (Python)
%o A014567 from math import gcd
%o A014567 from sympy import divisor_sigma
%o A014567 def ok(n): d = divisor_sigma(n, 1); return gcd(n, d) == 1
%o A014567 print([k for k in range(1, 134) if ok(k)]) # _Michael S. Branicky_, Mar 28 2022
%Y A014567 Cf. A003624.
%Y A014567 Cf. A069059 (complement).
%Y A014567 Includes A000961 as a subsequence.
%K A014567 nonn,easy,nice
%O A014567 1,2
%A A014567 _Eric W. Weisstein_
%E A014567 More terms from _Labos Elemer_