A036690 Product of a prime and the following number.

6, 12, 30, 56, 132, 182, 306, 380, 552, 870, 992, 1406, 1722, 1892, 2256, 2862, 3540, 3782, 4556, 5112, 5402, 6320, 6972, 8010, 9506, 10302, 10712, 11556, 11990, 12882, 16256, 17292, 18906, 19460, 22350, 22952, 24806, 26732, 28056, 30102

Offset: 1

Author

Felice Russo

Keywords

• nonn
• easy

Comments

The infinite sum over the reciprocals is given in A179119. - Wolfdieter Lang, Jul 10 2019

1/a(n) is the asymptotic density of numbers whose prime(n)-adic valuation is positive and even. - Amiram Eldar, Jan 23 2021

Examples

			a(3)=30 because prime(3)=5 and prime(3)+1=6, hence 5*6 = 30.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A036689, A034953, A065463, A179119, A306633.

Programs

Magma

[p^2+p: p in PrimesUpTo(250)]; // Vincenzo Librandi, Dec 19 2010

Mathematica

Table[(Prime[n] + 1) Prime[n], {n, 1, 100}] (* Artur Jasinski, Feb 06 2007 *)

PARI

a(n)=my(p=prime(n)); p*(p+1) \\ Charles R Greathouse IV, Mar 27 2020

Formula

a(n) = prime(n)*(prime(n)+1).

a(n) = A060800(n) - 1.

a(n) = 2*A034953(n). - Artur Jasinski, Feb 06 2007

From Amiram Eldar, Jan 23 2021: (Start)

Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(3) (A306633).

Product_{n>=1} (1 - 1/a(n)) = A065463. (End)

Sum_{n>=1} 1/a(n) = A179119. - R. J. Mathar, Mar 31 2025