A023195 Prime numbers that are the sum of the divisors of some n.

3, 7, 13, 31, 127, 307, 1093, 1723, 2801, 3541, 5113, 8011, 8191, 10303, 17293, 19531, 28057, 30103, 30941, 86143, 88741, 131071, 147073, 292561, 459007, 492103, 524287, 552793, 579883, 598303, 684757, 704761, 732541, 735307, 797161, 830833, 1191373

Offset

1,1

Comments

If n > 2 and sigma(n) is prime, then n must be an even power of a prime number. For example, 1093 = sigma(3^6). - T. D. Noe, Jan 20 2004

All primes of the form 2^n-1 (Mersenne primes) are in the sequence because if n is a natural number then sigma(2^(n-1)) = 2^n-1. So A000668 is a subsequence of this sequence. If sigma(n) is prime then n is of the form p^(q-1) where both p & q are prime (the proof is easy). - Farideh Firoozbakht, May 28 2005

Primes of the form 1 + p + p^2 + ... + p^k where p is prime.

If n = sigma(p^k) is in the sequence, then k+1 is prime. - Franklin T. Adams-Watters, Dec 19 2011

Primes that are a repunit in a prime base. - Franklin T. Adams-Watters, Dec 19 2011.

Except for 3, these primes are particular Brazilian primes belonging to A085104. These prime numbers are also Brazilian primes of the form (p^x - 1)/(p^y - 1), p prime, belonging to A003424, with here x is prime, and y = 1. [See section V.4 of Quadrature article in Links.] - Bernard Schott, Dec 25 2012

From Bernard Schott, Dec 25 2012: (Start)

Others subsequences of this sequence:

A053183 for 111_p = p^2 + p + 1 when p is prime.

A190527 for 11111_p = p^4 + p^3 + p^2 + p + 1 when p is prime.

A194257 for 1111111_p = p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime. (End)

Subsequence of primes from A002191. - Michel Marcus, Jun 10 2014

Links

David W. Wilson, Table of n, a(n) for n = 1..10000

Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Example

307 = 1 + 17 + 17^2; 307 and 17 are primes.

Mathematica

t={3}; lim=10^9; n=1; While[p=Prime[n]; k=2; s=1+p+p^2; s<lim, While[s<lim, If[PrimeQ[s], AppendTo[t, s]]; k=k+2; s=s+(1+p)p^(k-1)]; n++]; t=Union[t]
Select[DivisorSigma[1, Range[2 10^6]], PrimeQ]//Union (* Harvey P. Dale, Jun 18 2022 *)

Prog

(PARI) upto(lim)=my(v=List([3]), t); forprime(p=2, solve(x=1, lim^(1/4), x^4+x^3+x^2+x+1-lim), forprime(e=5, 1+log(lim)\log(p), if(isprime(t=sigma(p^(e-1))) && t<=lim, listput(v, t)))); forprime(p=2, solve(x=1, lim^(1/2), x^2+x+1-lim), if(isprime(t=p^2+p+1), listput(v, t))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Dec 20 2011
(Python)
from sympy import isprime, divisor_sigma
A023195_list = sorted(set([3]+[n for n in (divisor_sigma(d**2) for d in range(1, 10**4)) if isprime(n)])) # Chai Wah Wu, Jul 23 2016

Crossrefs

Intersection of A002191 and A000040.

Cf. A000203, A000668, A023194 (the n that produce these primes), A053696, A085104, A003424, A053183, A190527, A194257.

Sequence in context: A340870 A253850 A087578 * A100382 A292448 A222227

Adjacent sequences: A023192 A023193 A023194 * A023196 A023197 A023198

Keyword

nonn

Author

David W. Wilson

Status

approved