

A023195


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


17



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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^n1 (Mersenne primes) are in the sequence because if n is a natural number then sigma(2^(n1)) = 2^n1. So A000668 is a subsequence of this sequence. If sigma(n) is prime then n is of the form p^(q1) 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. AdamsWatters, Dec 19 2011
Primes that are a repunit in a prime base.  Franklin T. AdamsWatters, 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 brĂ©siliens, Quadrature, no. 76, avriljuin 2010, pages 3038; 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^(k1)]; n++]; t=Union[t]


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+1lim), forprime(e=5, 1+log(lim)\log(p), if(isprime(t=sigma(p^(e1)))&&t<=lim, listput(v, t)))); forprime(p=2, solve(x=1, lim^(1/2), x^2+x+1lim), 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

Cf. A000668, A023194 (the n that produce these primes), A053696, A085104, A003424, A053183, A190527, A194257.
Sequence in context: A069246 A253850 A087578 * A100382 A292448 A222227
Adjacent sequences: A023192 A023193 A023194 * A023196 A023197 A023198


KEYWORD

nonn


AUTHOR

David W. Wilson


STATUS

approved



