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

Author

Wesley Ivan Hurt, Apr 11 2015

Keywords

• nonn
• easy

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

Examples

			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
		

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

Crossrefs

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

Programs

Haskell

a256885 n = a000217 n - a000720 n  -- Reinhard Zumkeller, Apr 21 2015

Magma

[n*(n + 1)/2 - #PrimesUpTo(n): n in [1..60] ]; // Vincenzo Librandi, Apr 12 2015

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}]

PARI

vector(80, n, n*(n+1)/2 - primepi(n)) \\ Michel Marcus, Apr 13 2015

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)).

Extensions

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