A137441 Partial sums of partial sums of PrimePi(k).

0, 1, 4, 9, 17, 28, 43, 62, 85, 112, 144, 181, 224, 273, 328, 389, 457, 532, 615, 706, 805, 912, 1028, 1153, 1287, 1430, 1582, 1743, 1914, 2095, 2287, 2490, 2704, 2929, 3165, 3412, 3671, 3942, 4225, 4520, 4828, 5149, 5484, 5833, 6196, 6573, 6965, 7372, 7794

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Formula

a(n) = partial sum of A046992 = partial sum of partial sum of PrimePi(k) = partial sum of partial sum of A000720(k) = Sum_{j=1..n} (Sum_{k=1..j} PrimePi(k)).

a(n) = Sum_{i=1..n} (n+1-i)*pi(i), where pi = A000720. - Ridouane Oudra, Aug 31 2019

Maple

A000720 := proc(n) option remember ; numtheory[pi](n) ; end: A046992 := proc(n) option remember ; add( A000720(i), i=1..n) ; end: A137441 := proc(n) add( A046992(i), i=1..n) ; end: seq(A137441(n), n=1..80) ; # R. J. Mathar, Apr 23 2008
# second Maple program:
b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[pi](n+1), p[1]])(b(n-1)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=1..49);  # Alois P. Heinz, Oct 07 2021

Mathematica

Accumulate[Accumulate[PrimePi[Range[50]]]] (* Harvey P. Dale, Feb 17 2013 *)

Crossrefs

Cf. A000720, A046992.

Sequence in context: A008023 A008055 A301019 * A005744 A027367 A348238

Adjacent sequences: A137438 A137439 A137440 * A137442 A137443 A137444

Keyword

easy,nonn

Author

Jonathan Vos Post, Apr 17 2008

Extensions

More terms from R. J. Mathar, Apr 23 2008

Status

approved