A065358 - OEIS

%I #60 Apr 13 2024 14:57:08

%S 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,

%T 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,

%U 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

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

%C Partial sums of A065357.

%H N. J. A. Sloane, <a href="/A065358/b065358.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from Harry J. Smith).

%H Alberto Fraile, Roberto Martínez, and Daniel Fernández, <a href="https://arxiv.org/abs/1801.01540">Jacob's Ladder: Prime numbers in 2d</a>, 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]

%H Hans Havermann, <a href="/A065358/a065358.png">Graph of first 30 million terms</a>. [As can seen from A064940, one has to go out beyond 44 million terms to see any further runs of positive terms.]

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

%t Table[Sum[(-1)^(PrimePi[k]), {k,1,n}], {n,0,100}] (* _G. C. Greubel_, Feb 20 2018 *)

%t a[0] = 0; a[n_] := a[n] = a[n - 1] + (-1)^PrimePi[n]; Array[a, 105, 0] (* _Robert G. Wilson v_, Feb 20 2018 *)

%o (PARI) { a=0; for (n=1, 1000, a+=(-1)^primepi(n); write("b065358.txt", n, " ", a) ) } \\ _Harry J. Smith_, Sep 30 2009

%o [0] cat [(&+[(-1)^(#PrimesUpTo(k)):k in [1..n]]): n in [1..100]];  // _G. C. Greubel_, Feb 20 2018

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

%K easy,sign,nice

%O 0,5

%A _Jason Earls_, Oct 31 2001

%E Edited by _Frank Ellermann_, Feb 02 2002

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