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A023195 Prime numbers that are the sum of the divisors of some n. 15
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^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

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. - Bernard Schott, Dec 25 2012

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 with 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]

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

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

Sequence in context: A069246 A253850 A087578 * A100382 A222227 A152981

Adjacent sequences:  A023192 A023193 A023194 * A023196 A023197 A023198

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified June 22 23:15 EDT 2017. Contains 288633 sequences.