A064940 -id:A064940 - cp's OEIS frontend

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

Jason Earls, Oct 31 2001

Partial sums of A065357.

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.]

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

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

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 *)

{ 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

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)

A299300 Values of k such that A065358(k-1) = 0.

Original entry on oeis.org

1, 3, 7, 35, 39, 43, 51, 55, 79, 87, 91, 107, 111, 115, 835, 843, 1391, 1407, 1411, 1471, 1579, 1587, 1651, 1663, 1843, 1851, 3383, 3491, 3507, 3515, 3519, 3547, 3659, 3691, 3719, 3747, 3779, 3819, 3823, 3843, 3851, 3855, 3871, 3899, 3939, 3947, 3987, 3991

Offset: 1

N. J. A. Sloane, Feb 20 2018

nonn

Obtained by adding 1 to the terms of A064940.
Fraile et al. (2017) describe essentially the same sequence as A065358 except with offset 1 instead of 0. So the present sequence gives the values of k so that their version of the Jacob's Ladder sequence has the value 0.
For the first 7730 terms, see the b-file in A064940.

Alberto Fraile, Roberto Martínez, and Daniel Fernández, Jacob's Ladder: Prime numbers in 2d, arXiv preprint arXiv:1801.01540 [math.HO], 2017.

Cf. A065358, A064940.

A065358:= Table[Sum[(-1)^(PrimePi[k]), {k,1,n}], {n, 0, 500}]; Select[Range[50], A065358[[#]] == 0 &] (* G. C. Greubel, Feb 20 2018 *)

from sympy import nextprime
A299300_list, p, d, n, r = [], 2, -1, 0, False
while n <= 10**6:
    pn, k = p-n, d if r else -d
    if 0 < k <= pn:
        A299300_list.append(n+k)
    d += -pn if r else pn
    r, n, p = not r, p, nextprime(p) # Chai Wah Wu, Feb 21 2018

cp's OEIS Frontend

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

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