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A334813 Arithmetic numbers k (A003601) such that sigma(k)/d(k) is also an arithmetic number, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

%I A334813 #14 Jan 12 2024 01:28:53
%S A334813 1,5,6,11,13,14,15,20,29,37,38,39,41,43,44,45,49,53,54,56,57,59,60,61,
%T A334813 65,68,73,78,83,85,86,87,89,95,96,97,101,102,107,109,110,111,113,114,
%U A334813 116,118,123,125,129,131,134,135,137,139,142,143,145,147,150,153
%N A334813 Arithmetic numbers k (A003601) such that sigma(k)/d(k) is also an arithmetic number, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).
%C A334813 The number of terms not exceeding 10^k for k = 1, 2, ... is 3, 36, 426, 4744, 50442, 533806, 5585745, 58013810, 599272790, 6162302702, ... Apparently, this sequence has asymptotic density ~0.6.
%C A334813 Includes all the primes p such that (p+1)/2 is an odd prime, i.e., A005383 without the first term 3.
%C A334813 If p is in A240971 then p^2 is a term.
%H A334813 Amiram Eldar, <a href="/A334813/b334813.txt">Table of n, a(n) for n = 1..10000</a>
%e A334813 5 is a term since sigma(5)/d(5) = 6/2 = 3 is an integer, and so is sigma(3)/d(3) = 4/2 = 2.
%t A334813 rat[n_] := DivisorSigma[1, n]/DivisorSigma[0, n]; Select[Range[200], IntegerQ[(r = rat[#])] && IntegerQ[rat[r]] &]
%Y A334813 Cf. A000005, A000203, A003601, A102187.
%K A334813 nonn
%O A334813 1,2
%A A334813 _Amiram Eldar_, May 12 2020