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A023195
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Prime numbers that are the sum of the divisors of some n.
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6
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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; internal format)
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OFFSET
| 1,1
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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 (noe(AT)sspectra.com), 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 (mymontain(AT)yahoo.com), 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.
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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.
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38, included here with permission from the editors of Quadrature.
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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]
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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
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CROSSREFS
| Cf. A000668, A023194 (the n that produce these primes), A053696.
Sequence in context: A077314 A069246 A087578 * A100382 A152981 A112040
Adjacent sequences: A023192 A023193 A023194 * A023196 A023197 A023198
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KEYWORD
| nonn
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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