A002815 a(n) = n + Sum_{k=1..n} pi(k), where pi() = A000720. (Formerly M2523 N0996)

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, 487, 503, 519, 536, 553, 570, 587

Offset

0,3

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

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

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

Maple

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1) + 1 + numtheory[pi](n))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 01 2026

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 *)

Prog

(Haskell)
a002815 0 = 0
a002815 n = a046992 n + toInteger n  -- Reinhard Zumkeller, Feb 25 2012
(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
(PARI) a(n) = my(p=primes([0, n])); n + (n+1)*#p - vecsum(p); \\ Ruud H.G. van Tol, Feb 16 2024

Crossrefs

Cf. A000720, A046992.

Partial sums of A036234.

Sequence in context: A280944 A330321 A205726 * A342711 A109512 A025205

Adjacent sequences: A002812 A002813 A002814 * A002816 A002817 A002818

Keyword

nonn,nice,easy

Author

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

Status

approved