A256885 a(n) = n*(n + 1)/2 - pi(n), where pi(n) = A000720(n) is the prime counting function.

1, 2, 4, 8, 12, 18, 24, 32, 41, 51, 61, 73, 85, 99, 114, 130, 146, 164, 182, 202, 223, 245, 267, 291, 316, 342, 369, 397, 425, 455, 485, 517, 550, 584, 619, 655, 691, 729, 768, 808, 848, 890, 932, 976, 1021, 1067, 1113, 1161, 1210, 1260, 1311, 1363, 1415, 1469, 1524

Offset

1,2

Comments

Number of lattice points (x,y) in the region 1 <= x <= n, 1 <= y <= n - A010051(n); see example.

This sequence gives the row sums of the triangle A257232. - Wolfdieter Lang, Apr 21 2015

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime Counting Function

Wikipedia, Prime-counting function

Formula

a(n) = A000217(n) - A000720(n).

a(n) - a(n-1) = A014684(n), n >= 2.

a(n) = Sum_{i=1..n} A014684(i).

a(n) = 1 + Sum_{i=2..n}(i - A000720(i) + A000720(i-1)).

Example

10 .                             x
9  .                          x  x
8  .                       x  x  x
7  .                    .  x  x  x
6  .                 x  x  x  x  x
5  .              .  x  x  x  x  x
4  .           x  x  x  x  x  x  x
3  .        .  x  x  x  x  x  x  x
2  .     .  x  x  x  x  x  x  x  x
1  .  x  x  x  x  x  x  x  x  x  x
0  .__.__.__.__.__.__.__.__.__.__.
   0  1  2  3  4  5  6  7  8  9  10

Maple

with(numtheory)[pi]: A256885:=n->n*(n+1)/2-pi(n): seq(A256885(n), n=1..100);

Mathematica

Table[n (n + 1)/2 - PrimePi[n], {n, 1, 50}]

Prog

(Magma) [n*(n + 1)/2 - #PrimesUpTo(n): n in [1..60] ]; // Vincenzo Librandi, Apr 12 2015
(PARI) vector(80, n, n*(n+1)/2 - primepi(n)) \\ Michel Marcus, Apr 13 2015
(Haskell)
a256885 n = a000217 n - a000720 n  -- Reinhard Zumkeller, Apr 21 2015

Crossrefs

Cf. A000217, A000720, A010051, A014684, A113638, A257232.

Sequence in context: A100057 A007590 A080476 * A293495 A330130 A053799

Adjacent sequences: A256882 A256883 A256884 * A256886 A256887 A256888

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 11 2015

Extensions

Edited, following the hint by Reinhard Zumkeller to change the offset. - Wolfdieter Lang, Apr 22 2015

Status

approved