cp's OEIS Frontend

A337339 Denominator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).

%I A337339 #36 Dec 21 2024 22:30:09
%S A337339 1,5,13,41,25,113,61,365,313,221,85,1013,145,109,613,3281,181,2813,
%T A337339 265,1985,1513,761,421,9113,1201,1301,7813,377,481,5513,685,29525,
%U A337339 2113,1625,2965,25313,841,2381,3613,17861,925,13613,1105,6845,15313,3785,1405,82013,7321,10805,4513,11705,1741,70313,4141,8821,6613,865
%N A337339 Denominator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).
%C A337339 All terms are members of A007310, because all terms of A337336 and A337337 are.
%C A337339 No 1's after the initial one at a(1) => No quasiperfect numbers. See comments in A336700 & A337342.
%C A337339 If any quasiperfect numbers qp exist, they must occur also in A325311.
%C A337339 Question: Is there any reliable lower bound for this sequence? See A337340, A337341.
%C A337339 Duplicate values are rare, but at least two cases exist: a(21) = a(74) = 1513 and a(253) = a(554) = 71065. - _Antti Karttunen_, Jan 03 2024
%H A337339 Antti Karttunen, <a href="/A337339/b337339.txt">Table of n, a(n) for n = 1..8191</a>
%H A337339 Antti Karttunen, <a href="/A337339/a337339.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A337339 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A337339 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A337339 a(n) = A337336(n) / A337337(n) = A048673(n^2) / gcd(A048673(n^2), A336844(n^2)).
%F A337339 a(n) = A337336(n) / gcd(A337336(n), 1+A003973(n^2)).
%o A337339 (PARI)
%o A337339 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A337339 A337339(n) = { my(s=(A003961(n)^2),u=(s+1)/2); (u/gcd(1+sigma(s), u)); };
%o A337339 \\ Or alternatively as:
%o A337339 A337339(n) = { my(s=A003961(n^2)); denominator((1+sigma(s))/((s+1)/2)); };
%Y A337339 Cf. A003961, A003973, A007310, A048673, A074627, A074630, A209922, A324899, A325311, A336697, A336700, A336844, A337194, A337336, A337337, A337340, A337341, A337342.
%Y A337339 Cf. A337338 (numerators).
%Y A337339 Cf. also A336848, A336849.
%K A337339 nonn,frac
%O A337339 1,2
%A A337339 _Antti Karttunen_, Aug 24 2020