A064940 -id:A064940

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A065358

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

+10
12

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

(list; graph; refs; listen; history; text; internal format)

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).

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

A299300

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..48.

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

MATHEMATICA

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

PROG

(Python)

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

CROSSREFS

Cf. A065358, A064940.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2018

STATUS

approved

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