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A383428 Primitive terms in A066192: number k such that k is a term of A066192 and k/2 is not.

%I A383428 #10 Apr 27 2025 09:08:37
%S A383428 4,12,56,120,528,672,992,1456,2160,2208,6720,9024,9120,11904,13104,
%T A383428 16256,17472,24800,29568,55104,55552,73440,90816,95040,119040,120960,
%U A383428 121024,123648,131040,146688,151680,174720,195072,223104,297600,397440,399616,445536,505344
%N A383428 Primitive terms in A066192: number k such that k is a term of A066192 and k/2 is not.
%C A383428 If a(1) = 1 instead of 4, then this will be the sequence of primitive terms in A069519.
%C A383428 If k is a term then 2^m * k is a term in A066192 for all m >= 0.
%C A383428 If there is an odd term in this sequence it must be an odd perfect number (A000396). See the comments in A066192.
%C A383428 Except for 4, numbers k such that A091570(k) | k and k/A091570(k) is odd.
%H A383428 Amiram Eldar, <a href="/A383428/b383428.txt">Table of n, a(n) for n = 1..755</a> (terms below 10^11)
%H A383428 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>.
%t A383428 q[n_] := Module[{s = DivisorSigma[1, n/2^IntegerExponent[n, 2]] - If[OddQ[n], n, 0]}, Divisible[n, s] && OddQ[n/s]]; Select[Range[550000], # == 4 || (CompositeQ[#] && q[#]) &]
%o A383428 (PARI) isok(k) = if(k == 1 || isprime(k), 0, if(k == 4, 1, my(s = sigma(k >> valuation(k, 2)) - if(k%2, k)); !(k % s) && (k/s) % 2));
%Y A383428 Subsequence of A066191 and A066192.
%Y A383428 Cf. A000396, A069519, A091570.
%K A383428 nonn
%O A383428 1,1
%A A383428 _Amiram Eldar_, Apr 27 2025