

A249902


Numbers n such that 2n1 and sigma(n) are both primes.


3



2, 4, 9, 16, 64, 289, 1681, 2401, 3481, 4096, 15625, 65536, 85849, 262144, 491401, 531441, 552049, 683929, 703921, 734449, 1352569, 1885129, 3411409, 3892729, 5470921, 7091569, 7778521, 9247681, 10374841, 12652249, 18139081, 19439281, 22287841, 23902321
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OFFSET

1,1


COMMENTS

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 = kth Mersenne prime A000668(k) for k>=1.
Conjecture: 2 and 9 are the only numbers n such that 2n  1, 2n + 1 and sigma(n) are all primes.


LINKS



EXAMPLE

289 is in the sequence because 2*289  1 = 577 and sigma(289) = 307 (both primes).


MATHEMATICA

Select[Range[10^7], PrimeQ[2 #  1] && PrimeQ[DivisorSigma[1, #]] &] (* Vincenzo Librandi, Nov 15 2014 *)


PROG

(Magma) [n: n in [2..10000000]  IsPrime(2*n1) and IsPrime(SumOfDivisors(n))];
(PARI) for(n=1, 10^6, if(isprime(2*n1)&&isprime(sigma(n)), print1(n, ", "))) \\ Derek Orr, Nov 14 2014
(Python)
from sympy import isprime, divisor_sigma
A249902_list = [2]+[n for n in (d**2 for d in range(1, 10**3)) if isprime(2*n1) and isprime(divisor_sigma(n))] # Chai Wah Wu, Jul 23 2016


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



