A128930 a(n) = prime(n) * pi(n).

0, 3, 10, 14, 33, 39, 68, 76, 92, 116, 155, 185, 246, 258, 282, 318, 413, 427, 536, 568, 584, 632, 747, 801, 873, 909, 927, 963, 1090, 1130, 1397, 1441, 1507, 1529, 1639, 1661, 1884, 1956, 2004, 2076, 2327, 2353, 2674, 2702, 2758, 2786, 3165, 3345, 3405

Offset: 1

Author

Cino Hilliard, Apr 23 2007

Keywords

• easy
• nonn

Comments

Pi(n) = number of prime numbers <= n (A000720). Prime(n) = A000040(n).

Conjecture: For each n there is at least one prime p such that a(n) < p < a(n+1). From the conjecture follows that the prime gaps g(n) = p(n+1) - p(n) = O(sqrt(p(n))/log(p(n))). Legendre's hypothesis is that g(n) = O(sqrt(p(n))). - Thomas Ordowski, Aug 11 2012

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Wikipedia, Legendre's conjecture

Crossrefs

Cf. A000040, A000720, A128913.

Programs

Mathematica

Table[Prime[n] * PrimePi[n], {n, 50}] (* Harvey P. Dale, Mar 17 2011 *)

PARI

g(n) = for(x=1,n,y=prime(x)*primepi(x);print1(y","))

Formula

a(n) ~ (n log n)*(n/log n) = n^2. a(n) > n^2 for n > 4. - Thomas Ordowski, Aug 09 2012