A062357 a(n) = n*p(n+1)-(n+1)*p(n) = n*d(n)-p(n), where p(n) is the n-th prime and d(n) is the n-th prime-difference, A001223(n).

-1, 1, 1, 9, -1, 11, -3, 13, 31, -9, 35, 11, -15, 13, 43, 43, -25, 47, 9, -31, 53, 9, 55, 103, 3, -49, 5, -51, 7, 307, -3, 61, -71, 201, -79, 65, 65, -11, 67, 67, -97, 239, -105, -17, -107, 353, 353, -31, -129, -29, 73, -135, 289, 73, 73, 73, -155, 77, -41, -161, 327, 575, -55, -183, -53, 607, 71, 343, -209, -69, 73, 217

Offset: 1

Author

Labos Elemer, Jul 13 2001

Keywords

• sign
• easy

Comments

A sequence based on the solution of the equation: 1+(1+n)*prime(n)/x-n*prime(n+1)/x=0 for x. This is an irrational rotation-like sequence: the sequence is similar to a Beatty sequence. - Roger L. Bagula, Jun 06 2002

Examples

			n = 10: a(10) = 10*31-11*29 = 310-319 = -9;
n = 54: a(54) = 54*257-55*251 = 13878-13805 = 73;
n = 55: a(55) = 55*263-56*257 = 14465-14392 = 73; consecutive terms are often equal to each other.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A001223, A062742, A062743.

Programs

Magma

[n*NthPrime(n + 1) - (n + 1)*NthPrime(n): n in [1..75]]; // Vincenzo Librandi, Jun 29 2018

Maple

seq(n*ithprime(n+1)-(n+1)*ithprime(n),n=1..80); # Muniru A Asiru, Jun 29 2018

Mathematica

Table[(Prime[w+1]-Prime[w])*w-Prime[w], {w, 1, 1024}]

PARI

a(n)={n*prime(n + 1) - (n + 1)*prime(n)} \\ Harry J. Smith, Aug 06 2009

Formula

a(n) = n*A000040(n+1) - (n+1)*A000040(n) = n*A001223(n) - A000040(n).