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

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

Offset

1,1

Comments

Intersection of A006254 and A023194.

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.

Conjecture: 2 and 9 are the only numbers n such that 2n - 1, 2n + 1 and sigma(n) are all primes.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10622

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*n-1) and IsPrime(SumOfDivisors(n))];
(PARI) for(n=1, 10^6, if(isprime(2*n-1)&&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*n-1) and isprime(divisor_sigma(n))] # Chai Wah Wu, Jul 23 2016

Crossrefs

Cf. A000203, A006254, A023194, A019279, A249903.

Sequence in context: A283059 A283081 A344265 * A060401 A063981 A131095

Adjacent sequences: A249899 A249900 A249901 * A249903 A249904 A249905

Keyword

nonn

Author

Jaroslav Krizek, Nov 14 2014

Status

approved