A094759 Least k <= n such that n*sigma(k) = k*sigma(n), where sigma(n) is the sum of divisors of n (A000203).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Author

Amarnath Murthy, May 30 2004

Keywords

• nonn

Comments

Conjecture: There are infinitely many terms such that a(n)A050973 has those n, A050972 has the a(n).

See A095301 for a version of A050973 that do not duplicate every n that has several smaller k of the same abundancy. - Jeppe Stig Nielsen, Jul 09 2015

That conjecture is an easy fact: Since, e.g., (6,28) is a friendly pair, then so is (6k,28k) for any multiplier k with gcd(42,k)=1. So any n=28k, where gcd(42,k)=1, satisfies a(n)A095301 does not have asymptotic density zero. - Jeppe Stig Nielsen, Jul 09 2015

This sequence is related to Theorem 1 on p. 173 of the Erdős link in the following way. For a given x, let us consider the set of integers such that a(n) <= x, which is equivalent to removing duplicates from the current sequence. This set would begin with: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, ... So this set has the same number of elements as the number of distinct terms numbers of the form sigma(n)/n with 1 <= n <=x. Then by Erdős, it is c1*x + o(x), with 6/Pi^2 < c1 < 1. With x = 10^7, we find c1 ~= 0.98208... - Michel Marcus, Jul 21 2015

a(n) is the least k which has the same abundancy index as n, that is, minimal k for which sigma(k)/k = sigma(n)/n. - Antti Karttunen, Jul 10 2019

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• P. Erdős, Remarks on number theory II: Some problems on the sigma function, Acta Arith., 5 (1959), 171-177.

Crossrefs

Cf. A000203, A050972, A050973, A094757, A094758, A326200.

Cf. A095301 for n such that a(n) < n.

Cf. A000396 (positions of 6's), A005820 (positions of 120's).

Programs

Maple

N:= 100: # to get a(1) to a(N)
for n from 1 to N do
   v:= numtheory:-sigma(n)/n;
   if not assigned(R[v]) then R[v]:= n fi;
   A[n]:= R[v];
od:
seq(A[n],n=1..N); # Robert Israel, Jul 21 2015

Mathematica

Table[Module[{k=1,sn=DivisorSigma[1,n]},While[n DivisorSigma[1,k]!=k*sn,k++];k],{n,80}] (* Harvey P. Dale, Aug 03 2025 *)

PARI

for(n=1,74,s=sigma(n);k=1;while(n*sigma(k)!=k*s,k++);print1(k,","));

Extensions

Edited and extended by Don Reble and Klaus Brockhaus, May 31 2004