A065357 -id:A065357 - 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)

A064940 Values of k for which A065358(k) is 0.

Original entry on oeis.org

0, 2, 6, 34, 38, 42, 50, 54, 78, 86, 90, 106, 110, 114, 834, 842, 1390, 1406, 1410, 1470, 1578, 1586, 1650, 1662, 1842, 1850, 3382, 3490, 3506, 3514, 3518, 3546, 3658, 3690, 3718, 3746, 3778, 3818, 3822, 3842, 3850, 3854, 3870, 3898, 3938, 3946, 3986, 3990

Offset: 1

Jason Earls, Oct 31 2001

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..11631 (first 151 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. [They describe essentially the same sequence as A065358 except with offset 1 instead of 0, so their values for this sequence, which are given in A299300, differ by 1. - _N. J. A. Sloane_, Feb 20 2018]
Hans Havermann, Graph of first 30 million terms of A065358.
Hans Havermann, A064940 organized by axis crossing. The last number in a line is the axis crosser. Any number preceding it is a bouncer.
Don Reble, Table of n, a(n) for n = 1..97863 (The last term in the file is a(97863) = 694777169210.)

Cf. A065357, A299300.

m:= -1:
t:= 0:
Res:= 0,2:
for i from 3 to 5*10^7 by 2 do
  if isprime(i) then m:= -m fi;
  t:= t+2*m;
  if t = 0 then Res:= Res, i+1 fi;
od:
Res; # Robert Israel, Feb 20 2018

A065358 := Table[Sum[(-1)^(PrimePi[k]), {k,1,n}], {n,0,500}]; Select[Range[300], A065358[[#]] == 0 &] - 1  (* G. C. Greubel, Feb 20 2018 *)
c = s = 0; k = 1; lst = {0}; While[k < 100000, c = Mod[c + Boole[PrimeQ[k]], 2]; s = s + (-1)^c; If[s == 0, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Feb 20 2018 *)

{ n=s=0; for (m=1, 10^9, s+=(-1)^primepi(m); if (s==0, write("b064940.txt", n++, " ", m); if (n==150, return)) ) } \\ Harry J. Smith, Sep 30 2009

from sympy import nextprime
A064940_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:
        A064940_list.append(n+k-1)
    d += -pn if r else pn
    r, n, p = not r, p, nextprime(p) # Chai Wah Wu, Feb 21 2018

Initial term 0 added by N. J. A. Sloane, Feb 20 2018

A373923 a(n) = Sum_{d|n} (-1)^pi(d).

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 2, 2, 3, 0, 0, 0, 2, 2, 2, 3, 0, 0, 2, 2, 4, 0, 0, 0, -1, 0, 2, 2, 2, 2, 0, 2, 0, -2, 0, -1, 2, 2, 4, 4, 0, 2, 2, 2, 4, 0, 0, 0, 1, -2, 0, 0, 2, 0, 0, 4, 4, 2, 0, 2, 2, 0, 6, 3, 2, 0, 0, -2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 6, 3, 0, 0, 0, -2, 0, 2, 2

Offset: 1

Wesley Ivan Hurt, Jun 22 2024

sign

Inverse Möbius transform of (-1)^pi(n) (A065357).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720 (pi), A065357, A065358 (Jacob's Ladder sequence).

f:= proc(n) local d; add((-1)^numtheory:-pi(d),d=numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Sep 13 2024

Table[DivisorSum[n, (-1)^PrimePi[#] &], {n, 100}]

A373924 a(n) = Sum_{d|n} (-1)^pi(d) * mu(n/d).

Original entry on oeis.org

1, -2, 0, 2, -2, 0, 0, 0, 0, 4, -2, -2, 0, 2, 2, 0, -2, 0, 0, -2, 0, 4, -2, 0, 0, 0, -2, -4, 0, -2, -2, -2, 0, 2, 0, 0, 0, 2, 0, 0, -2, -2, 0, -2, 0, 4, -2, 0, -2, -2, 0, -2, 0, 4, 4, 2, 0, 2, -2, 0, 0, 4, 0, 2, 2, 0, -2, -2, 0, -4, 0, 2, -2, 0, -2, -4, 0, 0, 0, 0, 2

Offset: 1

Wesley Ivan Hurt, Jun 22 2024

sign

Möbius transform of (-1)^pi(n) (A065357).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720 (pi), A008683 (mu), A065357, A373923.

f:= proc(n) local d; add((-1)^numtheory:-pi(d)*numtheory:-mobius(n/d),d=numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Sep 13 2024

Table[DivisorSum[n, (-1)^PrimePi[#] MoebiusMu[n/#] &], {n, 100}]

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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A064940 Values of k for which A065358(k) is 0.

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A373923 a(n) = Sum_{d|n} (-1)^pi(d).

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A373924 a(n) = Sum_{d|n} (-1)^pi(d) * mu(n/d).

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Maple

Mathematica