A191225 Number of Ramanujan primes R_k between triangular numbers T(n-1) < R_k <= T(n).

0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 2, 1, 2, 0, 2, 3, 2, 1, 2, 2, 2, 1, 4, 2, 2, 1, 0, 4, 3, 5, 1, 3, 2, 1, 5, 1, 2, 3, 4, 4, 4, 2, 2, 2, 4, 2, 3, 4, 3, 5, 4, 3, 2, 5, 4, 2, 5, 1, 6, 1, 5, 5, 7, 2, 2, 1, 10, 6, 6, 2, 2, 5, 0, 3, 7, 5, 4, 6, 7, 4

Offset: 1

Author

John W. Nicholson, May 27 2011

Keywords

• nonn

Comments

The function eta(x), A191228, returns the greatest value of k of R_k <= x, and where R_k is the k-th Ramanujan prime (A104272).

Examples

			Write the numbers 1, 2, ... in a triangle with n terms in the n-th row; a(n) = number of Ramanujan primes in n-th row.
Triangle begins
1                 (0 Ramanujan primes, eta(1) = 0)
2  3              (1 Ramanujan primes, eta(3) - eta(1) = 1)
4  5  6           (0 Ramanujan primes, eta(6) - eta(3) = 0)
7  8  9  10       (0 Ramanujan primes, eta(10) - eta(6) = 0)
11 12 13 14 15    (1 Ramanujan primes, eta(15) - eta(10) = 1)
16 17 18 19 20 21 (1 Ramanujan primes, eta(21) - eta(15) = 1)
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Crossrefs

Cf. A000217, A000040, A104272, A191228, A014085, A190661, A083382, A191226, A191227.

Programs

Mathematica

terms = 100; nn = terms^2; R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s+1]] = k], {k, Prime[3 nn]}];
A104272 = R + 1;
eta = Table[Boole[MemberQ[A104272, k]], {k, 1, nn}] // Accumulate;
T[n_] := n(n+1)/2;
a[1] = 0; a[n_] := eta[[T[n]]] - eta[[T[n-1]]];
Array[a, terms] (* Jean-François Alcover, Nov 07 2018, using T. D. Noe's code for A104272 *)

Perl

use ntheory ":all"; sub a191225 { my $n = shift; ramanujan_prime_count( (($n-1)*$n)/2+1, ($n*($n+1))/2 ); } say a191225($) for 1..10; # _Dana Jacobsen, Dec 30 2015

Formula

a(n) = eta(T(n))- eta(T(n-1)).