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

Author

Jonathan Vos Post, Apr 17 2008

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A000720, A046992.

Programs

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 *)

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

Extensions

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