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

%I A002815 M2523 N0996 #63 Feb 17 2024 08:17:32
%S A002815 0,1,3,6,9,13,17,22,27,32,37,43,49,56,63,70,77,85,93,102,111,120,129,
%T A002815 139,149,159,169,179,189,200,211,223,235,247,259,271,283,296,309,322,
%U A002815 335,349,363,378,393,408,423,439,455,471
%N A002815 a(n) = n + Sum_{k=1..n} pi(k), where pi() = A000720.
%D A002815 H. Brocard, Reply to Query 1421, Nombres premiers dans une suite de différences, L'Intermédiaire des Mathématiciens, 7 (1900), 135-137.
%D A002815 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002815 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002815 T. D. Noe, <a href="/A002815/b002815.txt">Table of n, a(n) for n = 0..1000</a>
%F A002815 a(n) = A046992(n) + n for n > 0. - _Reinhard Zumkeller_, Feb 25 2012
%F A002815 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
%t A002815 Table[n + Sum[PrimePi[k], {k, 1, n}], {n, 0, 50}]
%t A002815 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 *)
%o A002815 (Haskell)
%o A002815 a002815 0 = 0
%o A002815 a002815 n = a046992 n + toInteger n  -- _Reinhard Zumkeller_, Feb 25 2012
%o A002815 (Python)
%o A002815 from sympy import primerange
%o A002815 def A002815(n): return n+(n+1)*len(p:=list(primerange(n+1)))-sum(p) # _Chai Wah Wu_, Jan 01 2024
%o A002815 (PARI) a(n) = my(p=primes([0,n])); n + (n+1)*#p - vecsum(p); \\ _Ruud H.G. van Tol_, Feb 16 2024
%Y A002815 Cf. A000720, A046992.
%K A002815 nonn,nice,easy
%O A002815 0,3
%A A002815 _N. J. A. Sloane_, _Robert G. Wilson v_, _Mira Bernstein_