cp's OEIS Frontend

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

%I A094759 #37 Aug 03 2025 15:02:23
%S A094759 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,
%T A094759 27,6,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,
%U A094759 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
%N A094759 Least k <= n such that n*sigma(k) = k*sigma(n), where sigma(n) is the sum of divisors of n (A000203).
%C A094759 Conjecture: There are infinitely many terms such that a(n)<n. A050973 has those n, A050972 has the a(n).
%C A094759 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
%C A094759 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)<n. This even shows that A095301 does not have asymptotic density zero. - _Jeppe Stig Nielsen_, Jul 09 2015
%C A094759 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
%C A094759 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
%H A094759 Robert Israel, <a href="/A094759/b094759.txt">Table of n, a(n) for n = 1..10000</a>
%H A094759 P. Erdős, <a href="https://www.renyi.hu/~p_erdos/1959-21.pdf">Remarks on number theory II: Some problems on the sigma function</a>, Acta Arith., 5 (1959), 171-177.
%p A094759 N:= 100: # to get a(1) to a(N)
%p A094759 for n from 1 to N do
%p A094759    v:= numtheory:-sigma(n)/n;
%p A094759    if not assigned(R[v]) then R[v]:= n fi;
%p A094759    A[n]:= R[v];
%p A094759 od:
%p A094759 seq(A[n],n=1..N); # _Robert Israel_, Jul 21 2015
%t A094759 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 *)
%o A094759 (PARI) for(n=1,74,s=sigma(n);k=1;while(n*sigma(k)!=k*s,k++);print1(k,","));
%Y A094759 Cf. A000203, A050972, A050973, A094757, A094758, A326200.
%Y A094759 Cf. A095301 for n such that a(n) < n.
%Y A094759 Cf. A000396 (positions of 6's), A005820 (positions of 120's).
%K A094759 nonn
%O A094759 1,2
%A A094759 _Amarnath Murthy_, May 30 2004
%E A094759 Edited and extended by _Don Reble_ and _Klaus Brockhaus_, May 31 2004