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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.
0
1, 3, 5, 9, 15, 45, 135, 315, 945
OFFSET
1,2
COMMENTS
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.
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.
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.
LINKS
Joshua Zelinsky, On the small prime factors of a non-deficient number, arXiv:2005.12118 [math.NT], 2020-2022.
MATHEMATICA
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 *)
PROG
(Sage)
def h(n):
return (sigma(n, 1))/n
def hchecker(k):
s=0
for i in range(1, k):
j=2^i+1
a=h(j)
if a> s:
print(i)
s=a
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Joshua Zelinsky, May 13 2020
EXTENSIONS
More terms from Amiram Eldar, May 13 2020, using A069061.
STATUS
approved