cp's OEIS Frontend

A334876 Numbers m with the property that sigma(2^m+1)/(2^m+1) > sigma(2^k+1)/(2^k+1) for all k < m, where sigma is the sum of divisors function, A000203.

%I A334876 #28 Nov 22 2023 19:22:00
%S A334876 1,3,5,9,15,45,135,315,945
%N A334876 Numbers m with the property that sigma(2^m+1)/(2^m+1) > sigma(2^k+1)/(2^k+1) for all k < m, where sigma is the sum of divisors function, A000203.
%C A334876 Set h(m) = sigma(m)/m. Then the sequence lists the numbers m at which record values of h(2^m+1) occur. This sequence is essentially defined similarly to A004394 but restricted to looking just at numbers which are one more than a power of 2.
%C A334876 The sequence is infinite. This can be proved by seeing that we can make h(2^m+1) arbitrarily large. Note that if p is a prime which is 3 (mod 8), then p|2^m+1 for any odd m such that (p-1)/2|m. By the strong version of Dirichlet's theorem the sum of the reciprocals of the primes which are 3 (mod 8) diverges. So we can make h(2^m+1) arbitrarily large by taking m as the largest odd divisor of k! for large k.
%C A334876 It appears that every term in the sequence is odd. This seems likely to be true since if m is even then 2^m+1 is not divisible by 3, which makes it much harder to make h(2^m+1) large.
%H A334876 Joshua Zelinsky, <a href="https://arxiv.org/abs/2005.12118">On the small prime factors of a non-deficient number</a>, arXiv:2005.12118 [math.NT], 2020-2022.
%t A334876 r[n_] := DivisorSigma[1, 2^n+1]/(2^n + 1); seq = {}; rm = 1; Do[r1 = r[n]; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 50}]; seq (* _Amiram Eldar_, May 15 2020 *)
%o A334876 (Sage)
%o A334876 def h(n):
%o A334876     return (sigma(n,1))/n
%o A334876 def hchecker(k):
%o A334876     s=0
%o A334876     for i in range(1,k):
%o A334876         j=2^i+1
%o A334876         a=h(j)
%o A334876         if a> s:
%o A334876             print(i)
%o A334876             s=a
%Y A334876 Cf. A000203, A004394, A069061.
%K A334876 nonn,more
%O A334876 1,2
%A A334876 _Joshua Zelinsky_, May 13 2020
%E A334876 More terms from _Amiram Eldar_, May 13 2020, using A069061.