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A128913

a(n) = n*pi(n).

0, 2, 6, 8, 15, 18, 28, 32, 36, 40, 55, 60, 78, 84, 90, 96, 119, 126, 152, 160, 168, 176, 207, 216, 225, 234, 243, 252, 290, 300, 341, 352, 363, 374, 385, 396, 444, 456, 468, 480, 533, 546, 602, 616, 630, 644, 705, 720, 735, 750, 765, 780, 848, 864, 880, 896

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OFFSET

1,2

COMMENTS

Pi(n) = number of primes <= n (see A000720).

Conjecture: For each n there is at least one prime p such that 2*a(n) < p < 2*a(n+1). From the conjecture follows that the prime gaps g(n) = prime(n+1) - prime(n) = O(sqrt(prime(n)/log(prime(n)))). - Thomas Ordowski, Aug 12 2012

Number of primes that are obtained when listing all reduced fractions i/j with 1<=i,j<=n. - Michel Marcus, Sep 09 2015

LINKS

Table of n, a(n) for n=1..56.

FORMULA

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

a(n) ~ n^2/log n. - Thomas Ordowski, Aug 12 2012

G.f.: x*f'(x), where f(x) = Sum_{k>=1} x^prime(k)/(1 - x). - Ilya Gutkovskiy, Apr 10 2017

EXAMPLE

a(7) = 28 because there are four primes less than or equal to 7 (namely 2, 3, 5, 7) and 7 * 4 = 28.

MATHEMATICA

Table[n PrimePi[n], {n, 60}] (* Alonso del Arte, Aug 14 2012 *)

PROG