A131872 Set m = 0, n = 1. Find smallest k >= 2 such that pi(k) = (k-pi(k)) - (m-pi(m)); set a(n) = pi(k), m = k, n = n+1. Repeat.

1, 4, 8, 11, 16, 23, 30, 39, 50, 62, 78, 97, 119, 141, 172, 205, 242, 284, 334, 393, 455, 531, 615, 704, 811, 928, 1059, 1213, 1373, 1560, 1761, 1988, 2239, 2524, 2833, 3180, 3557, 3983, 4448, 4942, 5503, 6126, 6791, 7522, 8331, 9228, 10188, 11228

Offset: 1

Author

Manuel Valdivia, Oct 05 2007

Keywords

• nonn

Comments

For n>1, a(n)-a(n-1) is approximately pi(n)^2/n.

Examples

			m=0, n=1; pi(2) = (2-1)-(0) = 1 = number of nonprimes from 1 to 2, a(1) = 1 is a term. Now n=2, m=2.
pi(9) = (9-4)-(2-1) = 4 = number of nonprimes from 3 to 9, a(2) = 4 is a term. Now n=3, m=9.
pi(21) = (21-8)-(9-4) = 8 = number of nonprimes from 10 to 21, a(3) = 8 is a term.
		

Links

• A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
• Eric Weisstein's World of Mathematics, Prime Counting Function

Crossrefs

Cf. A000720, A062298.

Programs

Mathematica

m=0; Do[If[PrimePi[n]==(n-PrimePi[n])-(m-PrimePi[m]), Print[PrimePi[n]]; m=n], {n, 1, 10^6, 1}]

PARI

lista(nn) = my(m=0, list = List()); for (n=1, nn, my(k=2); while(primepi(k) != (k-primepi(k)) - (m-primepi(m)), k++); listput(list, primepi(k)); m = k;); Vec(list); \\ Michel Marcus, Nov 13 2023

Extensions

Edited by N. J. A. Sloane, Nov 05 2007