A308754 a(0) = 0, a(n) = a(n-1) + 1 if 2*n + 3 is prime, otherwise a(n) = a(n-1).

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 9, 9, 9, 10, 10, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 16, 17, 17, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 28, 28, 28, 28, 28, 28, 29, 29, 30, 30, 30, 31

Offset: 0

Author

Carlos Fernandez Rivero, Jun 22 2019

Keywords

• nonn
• easy

Comments

It appears that A000040(a(n)) ~ 2*n as n tends to infinity. (See Mar 12 2012 note from Vladimir Shevelev in A060308.)

Examples

			a(0) = 0 (by definition).
a(1) = 1 = a(0) + 1, because 2*1 + 3 is prime;
a(2) = 2 = a(1) + 1, because 2*2 + 3 is prime;
a(3) = 2 = a(2),     because 2*3 + 3 is not prime;
a(4) = 3 = a(3) + 1, because 2*4 + 3 is prime.
		

Links

• Carlos Fernandez Rivero, Table of n, a(n) for n = 0..10000
• Carlos Fernandez Rivero, Plots of a(n) for n = 0..200 and n = 0..10000

Crossrefs

Cf. A000040, A000720, A060265, A060308, A085090, A099801, A101264.

Programs

BASIC

' p(n) contains the prime sequence except for 2. p(0)=3
' output in the a(n) sequence for 0 <= n <= maxterm
ip = -1
For n = 0 To maxterm
   If (2 * n + 3) = p(ip+1) Then
      ip = ip + 1
   End If
   a(n) = ip
Next n

Magma

[#PrimesUpTo(2*n + 4) - 2: n in [0..80] ]; // Vincenzo Librandi, Aug 01 2019

Mathematica

a[0] = 0; a[n_] := a[n] = a[n - 1] + Boole@PrimeQ[2 n + 3]; Array[a, 100, 0] (* Amiram Eldar, Jul 06 2019 *)

Formula

a(n) = a(n-1) + A101264(n+1), n > 0.

a(n) = A000720(2 * (n+2)) - 2.

a(n) = A099801(n+1) - 2.

a(n) = n - A210469(n+2).

A000040(a(n) + 2) = A060265(n+2).

A000040(a(n) + 2) = A060308(n+2).

A000040(a(n) + 2) = A085090(n+2), if 2*n + 3 is prime, otherwise 0.