A191225 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

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).

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

FORMULA

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

EXAMPLE

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)

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A126043 A360170 A306659 * A223893 A358009 A358010

Adjacent sequences: A191222 A191223 A191224 * A191226 A191227 A191228

KEYWORD

nonn

AUTHOR

John W. Nicholson, May 27 2011

STATUS

approved