A247938 Sum of divisors of 2^prime(n)-1.

4, 8, 32, 128, 2160, 8192, 131072, 524288, 8567136, 539922240, 2147483648, 138055271872, 2199187780272, 8817412930560, 140828559963840, 9008745449302368, 576463955735383776, 2305843009213693952, 147573953351708377936, 2361193635521975063040

Offset: 1

Author

Vincenzo Librandi, Sep 27 2014

Keywords

• nonn

Comments

b-file computed with factorizations in Wagstaff link. a(167) corresponding to 2^991-1 is currently the first unknown term. - Jens Kruse Andersen, Sep 28 2014

Conjecture: a(n)/2^prime(n) reaches its maximum value 135/128 at n = 5. - Jianing Song, Dec 31 2022

References

• R. Bojanić, Asymptotic evaluations of the sum of divisors of certain numbers (in Serbo-Croatian), Bull. Soc. Math.-Phys, R.P. Macédoine, Vol. 5 (1954), pp. 5-15.
• József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 96.

Links

• Jens Kruse Andersen and Amiram Eldar, Table of n, a(n) for n = 1..197 (terms 1..166 Jens Kruse Andersen)
• Paul Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, AMS, 2009, p. 271, exercise 22.
• Sam Wagstaff, The Cunningham Project.

Crossrefs

Subsequence of A075708.

Cf. A000203, A001348, A034785.

Programs

Magma

[SumOfDivisors(2^p-1): p in PrimesUpTo(100)];

Maple

with(numtheory): A247938:=n->sigma(2^ithprime(n)-1): seq(A247938(n), n=1..20); # Wesley Ivan Hurt, Sep 27 2014

Mathematica

Table[DivisorSigma[1, 2^Prime[n]-1], {n, 30}]

PARI

vector(50,n,sigma(2^prime(n)-1)) \\ Derek Orr, Sep 27 2014

Formula

a(n) = A000203(A001348(n)). - Michel Marcus, Sep 27 2014

Limit_{n->oo} a(n)/A001348(n) = 1 (Bojanić, 1954). - Amiram Eldar, Mar 04 2021