A230980 Number of primes <= n, starting at n=0.

0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21

Offset: 0

Author

Omar E. Pol, Nov 02 2013

Keywords

• nonn

Comments

Essentially identical to A000720, except that sequence, being an arithmetical sequence, starts at n = 1. - N. J. A. Sloane, Jun 21 2017

Also, on the first quadrant of the square grid, consider a diagram in which the number of cells in the horizontal bar of the k-th row is equal to the k-th prime, see example. The total length of the boundary segments between the structure formed by the first k horizontal bars and the structure formed by the vertical bars, from [0, 0], is equal to A014688(k). a(n) is the number of cells in the vertical bar of the n-th column.

Note that in a similar diagram for A000720 the lengths of the horizontal bars give A006093 (primes minus 1) not A000040 (the prime numbers) because A000720 has only one zero, not two.

Also, the number of distinct prime factors of the factorial number n!. - Torlach Rush, Jan 17 2014

The lengths of the boundary horizontal segments between the structure formed by the horizontal bars and the structure formed by the vertical bars of the diagram gives A054541. The zig-zag path formed by the boundary segments is in A230850. - Omar E. Pol, Jun 22 2017

Examples

			Illustration of initial terms:
.     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
31   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
29   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
23   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
19   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
17   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
13   |_ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
11   |_ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
7    |_ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
5    |_ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
3    |_ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
2    |_ _|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
n:    0 1 2 3 4 5 6 7 8 9...
a(n): 0 0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000040, A000720, A001223, A006093, A007504, A014688, A054541, A141042, A182986, A230850.

Partial sums of A010051.

Programs

Magma

[0] cat [#PrimesUpTo(n-1): n in [2..200] ]; // Vincenzo Librandi, Jun 22 2017

Mathematica

Array[PrimePi@# &, 90, 0] (* Robert G. Wilson v, Jun 21 2017 *)
Accumulate[Table[If[PrimeQ[n],1,0],{n,0,100}]] (* Harvey P. Dale, Mar 16 2023 *)

PARI

a(n)=primepi(n) \\ Charles R Greathouse IV, Jun 23 2017

Extensions

Offset and definition changed by N. J. A. Sloane, Jun 21 2017