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%I #66 Jun 18 2022 17:35:48
%S 3,7,13,31,127,307,1093,1723,2801,3541,5113,8011,8191,10303,17293,
%T 19531,28057,30103,30941,86143,88741,131071,147073,292561,459007,
%U 492103,524287,552793,579883,598303,684757,704761,732541,735307,797161,830833,1191373
%N Prime numbers that are the sum of the divisors of some n.
%C 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
%C 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
%C Primes of the form 1 + p + p^2 + ... + p^k where p is prime.
%C If n = sigma(p^k) is in the sequence, then k+1 is prime. - _Franklin T. Adams-Watters_, Dec 19 2011
%C Primes that are a repunit in a prime base. - _Franklin T. Adams-Watters_, Dec 19 2011.
%C 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
%C From _Bernard Schott_, Dec 25 2012: (Start)
%C Others subsequences of this sequence:
%C A053183 for 111_p = p^2 + p + 1 when p is prime.
%C A190527 for 11111_p = p^4 + p^3 + p^2 + p + 1 when p is prime.
%C A194257 for 1111111_p = p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime. (End)
%C Subsequence of primes from A002191. - _Michel Marcus_, Jun 10 2014
%H David W. Wilson, <a href="/A023195/b023195.txt">Table of n, a(n) for n = 1..10000</a>
%H Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
%e 307 = 1 + 17 + 17^2; 307 and 17 are primes.
%t 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]
%t Select[DivisorSigma[1,Range[2 10^6]],PrimeQ]//Union (* _Harvey P. Dale_, Jun 18 2022 *)
%o (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
%o (Python)
%o from sympy import isprime, divisor_sigma
%o 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
%Y Intersection of A002191 and A000040.
%Y Cf. A000203, A000668, A023194 (the n that produce these primes), A053696, A085104, A003424, A053183, A190527, A194257.
%K nonn
%O 1,1
%A _David W. Wilson_
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