

A065358


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


11



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.)
Theophilus Agama, The prime index function, arXiv:1905.03112 [math.GM], 2019.
Alberto Fraile, Roberto MartÃnez, and Daniel FernÃ¡ndez, 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/22978747/25/1/5 [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: A053646 A080776 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



