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
Links
- N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (first 1000 terms from Harry J. Smith).
- Alberto Fraile, Roberto MartÃnez, 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.]
Programs
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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
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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 *) -
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
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)
Comments