A066938 Primes of the form p*q+p+q, where p and q are primes.

11, 17, 23, 31, 41, 47, 53, 59, 71, 79, 83, 89, 107, 113, 127, 131, 151, 167, 179, 191, 227, 239, 251, 263, 269, 271, 293, 311, 359, 383, 419, 431, 439, 443, 449, 479, 491, 503, 521, 587, 593, 599, 607, 631, 647, 659, 683, 701, 719, 727, 743, 773, 809, 827

Offset: 1

Author

Reinhard Zumkeller, Jan 24 2002

Keywords

• nonn

Comments

For p not equal to q, either p*q or p+q is odd, so their sum is odd.

The representation is ambiguous, e.g. 2*7+2+7 = 23 = 3*5+3+5.

Complement of A198273 with respect to A000040. - Reinhard Zumkeller, Oct 23 2011

None of these primes are in A158913 since if p*q+p+q is a prime, then sigma(p*q+p+q) = sigma(p*q). - Amiram Eldar, Nov 15 2021

Examples

			59 is in the sequence because 59 = 2 * 19 + 2 + 19.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A054973, A067432, A072668, A072673, A088709, A158913, A198273, A198277.

Programs

Haskell

a066938 n = a066938_list !! (n-1)
a066938_list = map a000040 $ filter ((> 0) . a067432) [1..]
-- Reinhard Zumkeller, Oct 23 2011

Mathematica

nn = 1000; n2 = PrimePi[nn/3]; Select[Union[Flatten[Table[(Prime[i] + 1) (Prime[j] + 1) - 1, {i, n2}, {j, n2}]]], # <= nn && PrimeQ[#] &]

PARI

is(n)=fordiv(n+1,d,my(p=d-1,q=(n+1)/d-1); if(isprime(p) && isprime(q), return(isprime(n)))); 0 \\ Charles R Greathouse IV, Jul 23 2013

Formula

A067432(A049084(a(n))) > 0. - Reinhard Zumkeller, Oct 23 2011

A054973(a(n)+1) >= 2. - Amiram Eldar, Nov 15 2021

Extensions

Edited by Robert G. Wilson v, Feb 01 2002