A249903 Numbers n such that 2n+1 and sigma(n) are both noncomposite numbers.

1, 2, 9, 729

Offset: 1

Author

Jaroslav Krizek, Nov 14 2014

Keywords

• nonn
• more

Comments

If a(5) exists, it must be a square bigger than 3*10^8.

Intersection of A005097 and A023194.

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

From Hiroaki Yamanouchi, Nov 19 2014: (Start)

a(n) (n >= 3) must be of the form 3^(2k) for some positive integer k.

a(5) (if it exists) >= 3^877000 (see A003306 and A028491).

(End)

Examples

			Number 729 is in the sequence because 2*729 + 1 = 1459 and sigma(729) = 1093 (both primes).
		

Crossrefs

Cf. A000203, A003306, A005097, A023194, A028491, A249902.

Programs

Magma

[1] cat [n: n in [1..10000000] | IsPrime(2*n+1) and IsPrime(SumOfDivisors(n))]; // corrected by Vincenzo Librandi, Nov 14 2014

Mathematica

Join[{1}, Select[Range[0, 1000], PrimeQ[DivisorSigma[1, #]]&& PrimeQ[2 # + 1] &]] (* Vincenzo Librandi, Nov 14 2014 *)
Join[{1},Select[Range[1000],AllTrue[{2#+1,DivisorSigma[1,#]},PrimeQ]&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 06 2019 *)