A128930 -id:A128930 - cp's OEIS frontend

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

Offset: 1

Cino Hilliard, Apr 23 2007

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

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

Cf. A000720, A128930.

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

PARI
```
a(n) = n*primepi(n);
```

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

A217254 a(n) = round(primepi(n) * prime(n)/n).

Original entry on oeis.org

0, 2, 3, 4, 7, 7, 10, 10, 10, 12, 14, 15, 19, 18, 19, 20, 24, 24, 28, 28, 28, 29, 32, 33, 35, 35, 34, 34, 38, 38, 45, 45, 46, 45, 47, 46, 51, 51, 51, 52, 57, 56, 62, 61, 61, 61, 67, 70, 69, 69, 69, 69, 73, 74, 75, 75, 76, 75, 80, 80, 84, 85, 88, 87, 87, 86, 94

Offset: 1

Alexander R. Povolotsky, Mar 16 2013

For n < 10^7, a(n-1) > a(n) happens only for n composite. For n < 10^8, a(n-1) - a(n) <= 2. On the contrary, a(n) - a(n-1) seems to grow slowly and up to 10^5, 10^6, 10^7 and 10^8 is equal to 21, 26, 30, and 34, respectively. - Giovanni Resta, Mar 21 2013

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A128930.

Table[Floor[PrimePi[n]*Prime[n]/n + 1/2], {n, 100}] (* T. D. Noe, Mar 20 2013 *)

a(n)=prime(n)*primepi(n)\/n \\ Charles R Greathouse IV, Mar 19 2013

a(n) ~ n. More specifically, a(n) = n + n log log n/log n + 2n log log n/log^2 n + O(n/log^2 n); the O-constant is between -1/2 and -3/2 for large n. - Charles R Greathouse IV, Mar 19 2013

cp's OEIS Frontend

A128913 a(n) = n*pi(n).

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