A065358 The Jacob's Ladder sequence: a(n) = Sum_{k=1..n} (-1)^pi(k), where pi = A000720.

0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 3, 2, 3, 4, 5, 6, 5, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 3, 4, 3, 2, 1, 0, -1, -2, -1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, -1, -2, -1, 0, 1, 2, 3, 4, 3, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, -1, -2, -1, 0, 1, 2, 3, 4, 5, 6, 5, 4

Offset

0,5

Comments

Partial sums of A065357.

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (first 1000 terms from Harry J. Smith).

Alberto Fraile, Roberto Martínez, and Daniel Fernández, Jacob's Ladder: Prime numbers in 2d, arXiv preprint arXiv:1801.01540 [math.HO], 2017. Also Prime Numbers in 2D, Math. Comput. Appl. 2020, 25, 5; https://www.mdpi.com/2297-8747/25/1/5 [They describe essentially this sequence except with offset 1 instead of 0 - N. J. A. Sloane, Feb 20 2018]

Hans Havermann, Graph of first 30 million terms. [As can seen from A064940, one has to go out beyond 44 million terms to see any further runs of positive terms.]

Maple

with(numtheory): f:=n->add((-1)^pi(k), k=1..n); [seq(f(n), n=0..60)]; # N. J. A. Sloane, Feb 20 2018

Mathematica

Table[Sum[(-1)^(PrimePi[k]), {k, 1, n}], {n, 0, 100}] (* G. C. Greubel, Feb 20 2018 *)
a[0] = 0; a[n_] := a[n] = a[n - 1] + (-1)^PrimePi[n]; Array[a, 105, 0] (* Robert G. Wilson v, Feb 20 2018 *)

Prog

(PARI) { a=0; for (n=1, 1000, a+=(-1)^primepi(n); write("b065358.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 30 2009
[0] cat [(&+[(-1)^(#PrimesUpTo(k)):k in [1..n]]): n in [1..100]];  // G. C. Greubel, Feb 20 2018

Crossrefs

Cf. A000720, A065357, A064940 (the zero terms).

Sequence in context: A376813 A360659 A297158 * A062329 A022958 A023444

Adjacent sequences: A065355 A065356 A065357 * A065359 A065360 A065361

Keyword

easy,sign,nice

Author

Jason Earls, Oct 31 2001

Extensions

Edited by Frank Ellermann, Feb 02 2002

Edited by N. J. A. Sloane, Feb 20 2018 (added initial term a(0)=0, added name suggested by the Fraile et al. paper)

Status

approved