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A249902 Numbers n such that 2n-1 and sigma(n) are both primes.

%I A249902 #18 Sep 08 2022 08:46:10
%S A249902 2,4,9,16,64,289,1681,2401,3481,4096,15625,65536,85849,262144,491401,
%T A249902 531441,552049,683929,703921,734449,1352569,1885129,3411409,3892729,
%U A249902 5470921,7091569,7778521,9247681,10374841,12652249,18139081,19439281,22287841,23902321
%N A249902 Numbers n such that 2n-1 and sigma(n) are both primes.
%C A249902 Intersection of A006254 and A023194.
%C A249902 Sequence is a supersequence of the even superperfect numbers m_k (A061652 or even terms from A019279) because sigma(m_k) = 2*(m_k)-1 = k-th Mersenne prime A000668(k) for k>=1.
%C A249902 Conjecture: 2 and 9 are the only numbers n such that 2n - 1, 2n + 1 and sigma(n) are all primes.
%H A249902 Chai Wah Wu, <a href="/A249902/b249902.txt">Table of n, a(n) for n = 1..10622</a>
%e A249902 289 is in the sequence because 2*289 - 1 = 577 and sigma(289) = 307 (both primes).
%t A249902 Select[Range[10^7], PrimeQ[2 # - 1] && PrimeQ[DivisorSigma[1, #]] &] (* _Vincenzo Librandi_, Nov 15 2014 *)
%o A249902 (Magma) [n: n in [2..10000000] | IsPrime(2*n-1) and IsPrime(SumOfDivisors(n))];
%o A249902 (PARI) for(n=1,10^6,if(isprime(2*n-1)&&isprime(sigma(n)),print1(n,", "))) \\ _Derek Orr_, Nov 14 2014
%o A249902 (Python)
%o A249902 from sympy import isprime, divisor_sigma
%o A249902 A249902_list = [2]+[n for n in (d**2 for d in range(1,10**3)) if isprime(2*n-1) and isprime(divisor_sigma(n))] # _Chai Wah Wu_, Jul 23 2016
%Y A249902 Cf. A000203, A006254, A023194, A019279, A249903.
%K A249902 nonn
%O A249902 1,1
%A A249902 _Jaroslav Krizek_, Nov 14 2014