A128913 - OEIS

%I #27 Apr 10 2017 23:29:28

%S 0,2,6,8,15,18,28,32,36,40,55,60,78,84,90,96,119,126,152,160,168,176,

%T 207,216,225,234,243,252,290,300,341,352,363,374,385,396,444,456,468,

%U 480,533,546,602,616,630,644,705,720,735,750,765,780,848,864,880,896

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

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

%C 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

%C a(n) ~ n^2/log n. - _Thomas Ordowski_, Aug 12 2012

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

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

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

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

%t Table[n Pi[n], {n, 60}] (* _Alonso del Arte_, Aug 14 2012 *)

%o (PARI) g(n) = for(x=1,n,y=x*primepi(x);print1(y","))

%Y Cf. A000720, A128930.

%K easy,nonn

%O 1,2

%A _Cino Hilliard_, Apr 23 2007