A337339 - OEIS

%I #36 Dec 21 2024 22:30:09

%S 1,5,13,41,25,113,61,365,313,221,85,1013,145,109,613,3281,181,2813,

%T 265,1985,1513,761,421,9113,1201,1301,7813,377,481,5513,685,29525,

%U 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 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 All terms are members of A007310, because all terms of A337336 and A337337 are.

%C No 1's after the initial one at a(1) => No quasiperfect numbers. See comments in A336700 & A337342.

%C If any quasiperfect numbers qp exist, they must occur also in A325311.

%C Question: Is there any reliable lower bound for this sequence? See A337340, A337341.

%C 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 Antti Karttunen, <a href="/A337339/b337339.txt">Table of n, a(n) for n = 1..8191</a>

%H Antti Karttunen, <a href="/A337339/a337339.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = A337336(n) / A337337(n) = A048673(n^2) / gcd(A048673(n^2), A336844(n^2)).

%F a(n) = A337336(n) / gcd(A337336(n), 1+A003973(n^2)).

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A337339(n) = { my(s=(A003961(n)^2),u=(s+1)/2); (u/gcd(1+sigma(s), u)); };

%o \\ Or alternatively as:

%o A337339(n) = { my(s=A003961(n^2)); denominator((1+sigma(s))/((s+1)/2)); };

%Y Cf. A003961, A003973, A007310, A048673, A074627, A074630, A209922, A324899, A325311, A336697, A336700, A336844, A337194, A337336, A337337, A337340, A337341, A337342.

%Y Cf. A337338 (numerators).

%Y Cf. also A336848, A336849.

%K nonn,frac

%O 1,2

%A _Antti Karttunen_, Aug 24 2020