A214873 Primes p such that 2*p + 1 is also prime and p + 1 is a highly composite number (definition 1).

3, 5, 11, 23, 179, 239, 359, 719, 5039, 55439, 665279, 6486479, 32432399, 698377679, 735134399, 1102701599, 20951330399, 3212537327999, 149602080797769599, 299204161595539199, 2718551763981393634806325317503999

Offset: 1

Author

Arkadiusz Wesolowski, Jul 30 2012

Keywords

• nonn

Comments

An equivalent definition of this sequence: odd Sophie Germain primes that differ from a highly composite number by 1.

Intersection of A005384 (Sophie Germain primes) and A072828.

With the exception of 5, a subsequence of A002515 (Lucasian primes).

Except for first two terms, this is a subsequence of A054723.

Except for n = 2, 2*a(n) + 1 is a prime factor of A000225(a(n)) (i.e., 2*23 + 1 divides 2^23 - 1).

Conjecture: for n >= 5, GCD(A000032(a(n)), A000225(a(n))) = 2*a(n) + 1.

Examples

			23 is a term because both 23 and 47 are primes and also 24 is a highly composite number.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..25
• Wikipedia, Sophie Germain prime

Crossrefs

Cf. A054723.

Programs

Mathematica

lst = {}; a = 0; Do[b = DivisorSigma[0, n + 1]; If[b > a, a = b; If[PrimeQ[n] && PrimeQ[2*n + 1], AppendTo[lst, n]]], {n, 1, 10^6, 2}]; lst