A256956 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

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