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A065358
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The Jacob's Ladder sequence: a(n) = Sum_{k=1..n} (-1)^pi(k), where pi = A000720.
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11
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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
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OFFSET
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0,5
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COMMENTS
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LINKS
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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]
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MAPLE
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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
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,sign,nice,changed
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AUTHOR
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EXTENSIONS
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Edited by N. J. A. Sloane, Feb 20 2018 (added initial term a(0)=0, added name suggested by the Fraile et al. paper)
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STATUS
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approved
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