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

1,1

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..25

Wikipedia, Sophie Germain prime

Example

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

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

Crossrefs

Cf. A054723.

Sequence in context: A074892 A074874 A051439 * A330951 A018116 A259643

Adjacent sequences: A214870 A214871 A214872 * A214874 A214875 A214876

Keyword

nonn

Author

Arkadiusz Wesolowski, Jul 30 2012

Status

approved