A256956 a(n) = pi(n) * pi(n+1), where pi(n) is the number of primes <= n.

0, 2, 4, 6, 9, 12, 16, 16, 16, 20, 25, 30, 36, 36, 36, 42, 49, 56, 64, 64, 64, 72, 81, 81, 81, 81, 81, 90, 100, 110, 121, 121, 121, 121, 121, 132, 144, 144, 144, 156, 169, 182, 196, 196, 196, 210, 225, 225, 225, 225, 225, 240, 256, 256, 256, 256, 256, 272

Offset

1,2

Comments

For n>1, a(n) is a perfect square (A000290) if and only if pi(n) = pi(n+1) [i.e., when n+1 is composite], and is a pronic number (A002378) when pi(n) < pi(n+1) [when n+1 is prime].

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000720(n) * A000720(n+1).

Example

a(5) = 9; pi(5) * pi(6) = 3 * 3 = 9.
a(6) = 12; pi(6) * pi(7) = 3 * 4 = 12.

Maple

with(numtheory): A256956:=n->pi(n)*pi(n+1): seq(A256956(n), n=1..100);

Mathematica

Table[PrimePi[n]*PrimePi[n + 1], {n, 100}]

Prog

(PARI) vector(100, n, primepi(n)*primepi(n+1)) \\ Derek Orr, Apr 13 2015
(Magma) [ #PrimesUpTo(n) * #PrimesUpTo(n+1):  n in [1..80] ]; // Vincenzo Librandi, Apr 14 2015
(Scheme) (define (A256956 n) (* (A000720 n) (A000720 (+ 1 n)))) ;; Antti Karttunen, Apr 18 2015

Crossrefs

Cf. A000290, A000720, A002378.

Sequence in context: A180107 A375983 A135146 * A257637 A258027 A053096

Adjacent sequences: A256953 A256954 A256955 * A256957 A256958 A256959

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 13 2015

Extensions

Comment clarified by Antti Karttunen, Apr 18 2015

Status

approved