%I #25 Apr 28 2023 14:14:11
%S 0,2,4,6,9,12,16,16,16,20,25,30,36,36,36,42,49,56,64,64,64,72,81,81,
%T 81,81,81,90,100,110,121,121,121,121,121,132,144,144,144,156,169,182,
%U 196,196,196,210,225,225,225,225,225,240,256,256,256,256,256,272
%N a(n) = pi(n) * pi(n+1), where pi(n) is the number of primes <= n.
%C 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].
%H Antti Karttunen, <a href="/A256956/b256956.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A000720(n) * A000720(n+1).
%e a(5) = 9; pi(5) * pi(6) = 3 * 3 = 9.
%e a(6) = 12; pi(6) * pi(7) = 3 * 4 = 12.
%p with(numtheory): A256956:=n->pi(n)*pi(n+1): seq(A256956(n), n=1..100);
%t Table[PrimePi[n]*PrimePi[n + 1], {n, 100}]
%o (PARI) vector(100,n,primepi(n)*primepi(n+1)) \\ _Derek Orr_, Apr 13 2015
%o (Magma) [ #PrimesUpTo(n) * #PrimesUpTo(n+1): n in [1..80] ]; // _Vincenzo Librandi_, Apr 14 2015
%o (Scheme) (define (A256956 n) (* (A000720 n) (A000720 (+ 1 n)))) ;; _Antti Karttunen_, Apr 18 2015
%Y Cf. A000290, A000720, A002378.
%K nonn,easy
%O 1,2
%A _Wesley Ivan Hurt_, Apr 13 2015
%E Comment clarified by _Antti Karttunen_, Apr 18 2015