cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A247939 Sum of divisors of 2^prime(n)+1.

Original entry on oeis.org

6, 13, 48, 176, 2736, 10928, 174768, 699056, 11184816, 727960800, 2863311536, 183355069408, 2967356682528, 11728124029616, 188313058624992, 12121838249371488, 768906329487027264, 3074457345618258608, 196765296972010592800, 3148244377723715041632
Offset: 1

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Author

Vincenzo Librandi, Sep 27 2014

Keywords

Comments

b-file computed with factorizations in Wagstaff link. a(166) corresponding to 2^983+1 is currently the first unknown term. - Jens Kruse Andersen, Sep 28 2014

Crossrefs

Programs

  • Magma
    [SumOfDivisors(2^p+1): p in PrimesUpTo(100)];
    
  • Maple
    with(numtheory): A247939:=n->sigma(2^ithprime(n)+1): seq(A247939(n), n=1..20); # Wesley Ivan Hurt, Sep 27 2014
  • Mathematica
    Table[DivisorSigma[1, 2^Prime[n] + 1], {n, 1, 20}]
  • PARI
    vector(50,n,sigma(2^prime(n)+1)) \\ Derek Orr, Sep 27 2014

Formula

a(n) = A000203(A098640(n)). - Michel Marcus, Sep 27 2014

A268130 Primes p of the form sigma(2^k + 1) - 1 for some k >= 0.

Original entry on oeis.org

2, 3, 5, 17, 47, 83, 257, 1301, 65537, 174767, 5048231, 51322664447, 188313058624991, 4768522825659911, 3148244377723715041631, 211635519858089932125000235391, 906780938207203620571208267879698943, 6392739029893008727817055462596999999
Offset: 1

Views

Author

Jaroslav Krizek, Jan 26 2016

Keywords

Comments

Corresponding values of k: 0, 1, 2, 4, 5, 6, 8, 10, 16, 17, 22, 35, 47, 52, 71, 97, 119, 122, 124, 190, 300, ...
Fermat primes from A019434 are in the sequence.

Examples

			Prime 47 is a term because for k = 5, sigma(2^5+1) - 1 = sigma(33) - 1 = 47.
		

Crossrefs

Programs

  • Magma
    Set(Sort([SumOfDivisors(2^n+1)-1: n in [0..300] | IsPrime(SumOfDivisors(2^n+1)-1)]));
    
  • PARI
    lista(nn) = for (n=0, nn, if (isprime(p=sigma(2^n + 1) - 1), print1(p, ", "))); \\ Michel Marcus, Jan 27 2016

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.

Original entry on oeis.org

1, 3, 5, 9, 15, 45, 135, 315, 945
Offset: 1

Views

Author

Joshua Zelinsky, May 13 2020

Keywords

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.

Crossrefs

Programs

  • 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 *)
  • 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

Extensions

More terms from Amiram Eldar, May 13 2020, using A069061.
Previous Showing 11-13 of 13 results.