A002815 a(n) = n + Sum_{k=1..n} pi(k), where pi() = A000720.

0, 1, 3, 6, 9, 13, 17, 22, 27, 32, 37, 43, 49, 56, 63, 70, 77, 85, 93, 102, 111, 120, 129, 139, 149, 159, 169, 179, 189, 200, 211, 223, 235, 247, 259, 271, 283, 296, 309, 322, 335, 349, 363, 378, 393, 408, 423, 439, 455, 471

Offset: 0

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Keywords

• nonn
• nice
• easy

References

• H. Brocard, Reply to Query 1421, Nombres premiers dans une suite de différences, L'Intermédiaire des Mathématiciens, 7 (1900), 135-137.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000720, A046992.

Programs

Haskell

a002815 0 = 0
a002815 n = a046992 n + toInteger n  -- Reinhard Zumkeller, Feb 25 2012

Mathematica

Table[n + Sum[PrimePi[k], {k, 1, n}], {n, 0, 50}]
Module[{nn=50,pp},pp=Accumulate[PrimePi[Range[0,nn]]];Total/@ Thread[ {Range[ 0,nn],pp}]] (* This program is significantly faster than the program above. *) (* Harvey P. Dale, Jan 03 2013 *)

PARI

a(n) = my(p=primes([0,n])); n + (n+1)*#p - vecsum(p); \\ Ruud H.G. van Tol, Feb 16 2024

Python

from sympy import primerange
def A002815(n): return n+(n+1)*len(p:=list(primerange(n+1)))-sum(p) # Chai Wah Wu, Jan 01 2024

Formula

a(n) = A046992(n) + n for n > 0. - Reinhard Zumkeller, Feb 25 2012

Conjectured g.f.: (Sum_{N>=1} x^A008578(N))/(1-x)^2 = (x + x^2 + x^3 + x^5 + x^7 + x^11 + x^13 + ...)/(1-x)^2. - L. Edson Jeffery, Nov 25 2013