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A256885
a(n) = n*(n + 1)/2 - pi(n), where pi(n) = A000720(n) is the prime counting function.
4
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
Eric Weisstein's World of Mathematics, 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
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