A292378 -id:A292378 - cp's OEIS frontend

A038698 Excess of 4k-1 primes over 4k+1 primes, beginning with prime 2.

0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 2, 3, 4, 5, 6, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 3, 4, 3, 4, 5, 4, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 3, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 6, 5, 6, 5, 6, 5, 6

Offset: 1

Hans Havermann

a(n) < 0 for infinitely many values of n. - Benoit Cloitre, Jun 24 2002

First negative value is a(2946) = -1, which is for prime 26861. - David W. Wilson, Sep 27 2002

Stan Wagon, The Power of Visualization, Front Range Press, 1994, p. 2.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Vladimir Pletser, Chart for n=1..100000

Cf. A007350, A007351, A038691, A051024, A066520.
Cf. A112632 (race of 3k-1 and 3k+1 primes), A216057, A269364.
Cf. A156749 (sequence showing Chebyshev bias in prime races (mod 4)), A199547, A267097, A267098, A267107, A292378.
Cf. A163805, A212159.
List of primes p such that a(p) = 0 is A007351. List of primes p such that a(p) < 0 is A199547. List of primes p such that a(p) = -1 is A051025. List of integers k such that a(prime(k)) = -1 is A051024. - Ya-Ping Lu, Jan 18 2025

ans:=[0]; ct:=0; for n from 2 to 2000 do
p:=ithprime(n); if (p mod 4) = 3 then ct:=ct+1; else ct:=ct-1; fi;
ans:=[op(ans),ct]; od: ans; # N. J. A. Sloane, Jun 24 2016

FoldList[Plus, 0, Mod[Prime[Range[2,110]], 4] - 2]
Join[{0},Accumulate[If[Mod[#,4]==3,1,-1]&/@Prime[Range[2,110]]]] (* Harvey P. Dale, Apr 27 2013 *)

for(n=2,100,print1(sum(i=2,n,(-1)^((prime(i)+1)/2)),","))

from sympy import nextprime; a, p = 0, 2; R = [a]
for _ in range(2,88): p=nextprime(p); a += p%4-2; R.append(a)
print(*R, sep = ', ')  # Ya-Ping Lu, Jan 18 2025

a(n) = Sum_{k=2..n} (-1)^((prime(k)+1)/2). - Benoit Cloitre, Jun 24 2002
a(n) = (Sum_{k=1..n} prime(k) mod 4) - 2*n (assuming that x mod 4 > 0). - Thomas Ordowski, Sep 21 2012
From Antti Karttunen, Oct 01 2017: (Start)
a(n) = A267098(n) - A267097(n).
a(n) = A292378(A000040(n)).
(End)
From Ridouane Oudra, Nov 04 2024: (Start)
a(n) = Sum_{k=2..n} i^(prime(k)+1), where i is the imaginary unit.
a(n) = Sum_{k=2..n} sin(3*prime(k)*Pi/2).
a(n) = Sum_{k=2..n} A163805(prime(k)).
a(n) = Sum_{k=2..n} A212159(k). (End)
a(n) = a(n-1) + prime(n) (mod 4) - 2, n >= 2. - Ya-Ping Lu, Jan 18 2025

A292377 a(1) = 0, and for n > 1, a(n) = a(A252463(n)) + [n == 3 (mod 4)].

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 3, 1, 3, 2, 2, 0, 3, 0, 4, 1, 1, 3, 5, 1, 0, 3, 1, 2, 5, 2, 6, 0, 2, 3, 3, 0, 6, 4, 4, 1, 6, 1, 7, 3, 1, 5, 8, 1, 0, 0, 4, 3, 8, 1, 2, 2, 3, 5, 9, 2, 9, 6, 2, 0, 2, 2, 10, 3, 4, 3, 11, 0, 11, 6, 1, 4, 3, 4, 12, 1, 0, 6, 13, 1, 4, 7, 6, 3, 13, 1, 3, 5, 5, 8, 5, 1, 13, 0, 3, 0, 13, 4, 14, 3, 2

Offset: 1

Antti Karttunen, Sep 17 2017

nonn

For numbers > 1, iterate the map x -> A252463(x) which divides even numbers by 2 and shifts every prime in the prime factorization of odd n one index step towards smaller primes. a(n) counts the numbers of the form 4k+3 encountered until 1 has been reached. The count includes also n itself if it is of the form 4k+3 (A004767).

In other words, locate the node which contains n in binary tree A005940 and traverse from that node towards the root, counting all numbers of the form 4k+3 that occur on the path.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A005940, A061395, A064989, A252463, A292375, A292376, A292378, A292383.

a[1] = 0; a[n_] := a[n] = a[Which[n == 1, 1, EvenQ@ n, n/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n]] + Boole[Mod[n, 4] == 3]; Array[a, 105]

a(1) = 0, and for n > 1, a(n) = a(A252463(n)) + floor((n mod 4)/3).
Equivalently, a(2n) = a(n), and for odd numbers n > 1, a(n) = a(A064989(n)) + [n == 3 (mod 4)].
a(n) = A000120(A292383(n)).
Other identities. For n >= 1:
a(n) >= A292376(n).
a(A000040(n)) = A267098(n).
1 + a(n) - A292375(n) = A292378(n).
For n >= 2, a(n) + A292375(n) = A061395(n).

A292375 a(1) = 1, and for n > 1, a(n) = a(A252463(n)) + [n == 1 (mod 4)].

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 1, 1, 4, 2, 4, 2, 3, 2, 4, 1, 3, 3, 1, 2, 5, 1, 5, 1, 3, 4, 1, 2, 6, 4, 2, 2, 7, 3, 7, 2, 2, 4, 7, 1, 4, 3, 3, 3, 8, 1, 3, 2, 5, 5, 8, 1, 9, 5, 2, 1, 4, 3, 9, 4, 5, 1, 9, 2, 10, 6, 2, 4, 2, 2, 10, 2, 2, 7, 10, 3, 3, 7, 4, 2, 11, 2, 3, 4, 6, 7, 3, 1, 12, 4, 2, 3, 13, 3, 13, 3, 2

Offset: 1

Antti Karttunen, Sep 17 2017

nonn

For numbers > 1, iterate the map x -> A252463(x) which divides even numbers by 2, and shifts every prime in the prime factorization of odd n one index step towards smaller primes. a(n) counts the numbers of the form 4k+1 encountered until 1 has been reached, which is also included in the count. The count includes also n itself if it is of the form 4k+1 (A016813), thus a(1) = 1.

In other words, locate the node which contains n in binary tree A005940 and traverse from that node towards the root, counting all numbers of the form 4k+1 that occur on the path.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A001248, A005940, A061395, A064989, A252463, A292374, A292377, A292378, A292381, A292583.

a[1] = 1; a[n_] := a[n] = a[Which[n == 1, 1, EvenQ@n, n/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n]] + Boole[Mod[n, 4] == 1]; Array[a, 105] (* Michael De Vlieger, Sep 17 2017 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A292375(n) = if(1==n,n,if(!(n%2),A292375(n/2),(if(1==(n%4),1,0)+A292375(A064989(n)))));

;; With memoization-macro definec.
(definec (A292375 n) (if (= 1 n) 1 (+ (if (= 1 (modulo n 4)) 1 0) (A292375 (A252463 n)))))

a(1) = 1, a(2n) = a(n), and for odd numbers n > 1, a(n) = a(A064989(n)) + [n == 1 (mod 4)].
a(n) = A000120(A292381(n)).
Other identities and observations. For n >= 1:
a(n) >= A292374(n).
a(A000040(n))-1 = A267097(n).
1 + A292377(n) - a(n) = A292378(n).
For n >= 2, a(n) + A292377(n) = A061395(n).
From Antti Karttunen, Apr 22 2022: (Start)
For n >= 2, a(n^2) = A061395(n). [Because A292377(n^2) = 0]
For n >= 1, a(A001248(n)) = n. [See comments in A292583]
(End)

A292584 Compound filter: a(n) = P(A292583(n), A292585(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 2, 8, 5, 13, 2, 4, 8, 19, 5, 25, 13, 9, 2, 32, 4, 41, 8, 12, 19, 51, 5, 16, 25, 5, 13, 72, 9, 85, 2, 18, 32, 14, 4, 98, 41, 26, 8, 112, 12, 128, 19, 8, 51, 145, 5, 46, 16, 33, 25, 180, 5, 18, 13, 49, 72, 200, 9, 220, 85, 13, 2, 24, 18, 242, 32, 60, 14, 265, 4, 288, 98, 8, 41, 19, 26, 313, 8, 4, 112, 339, 12, 33, 128, 62, 19, 365, 8, 25, 51, 84, 145

Offset: 1

Antti Karttunen, Sep 20 2017

nonn

From Antti Karttunen, Sep 25 2017: (Start)
Some of the sequences this filter matches to:
For all i, j: a(i) = a(j) => A053866(i) = A053866(j).
For all i, j: a(i) = a(j) => A061395(i) = A061395(j). (also A006530, etc.)
For all i, j: a(i) = a(j) => A292378(i) = A292378(j).
The latter two implications follow simply because:
  A001222(A278222(A292383(n))) = A000120(A292383(n)) = A292377(n)
and, similarly, for n > 1,
  A001222(A278222(A292385(n))) = A000120(A292381(n)) = A292375(n),
and the sum of A292375(n) and A292377(n) is A061395(n) [index of the largest prime dividing n], while A292378 has been defined as 1 + their difference.
The case A053866 follows because of the component A292583, see comments under that entry. (End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A292383, A292385, A292583, A292585.

Cf. also A006530, A061395, A292378 (some of the matched sequences).

a(n) = (1/2)*(2 + ((A292583(n)+A292585(n))^2) - A292583(n) - 3*A292585(n)).

cp's OEIS Frontend

A038698 Excess of 4k-1 primes over 4k+1 primes, beginning with prime 2.

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A292377 a(1) = 0, and for n > 1, a(n) = a(A252463(n)) + [n == 3 (mod 4)].

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A292375 a(1) = 1, and for n > 1, a(n) = a(A252463(n)) + [n == 1 (mod 4)].

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Programs

Mathematica

PARI

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Formula

A292584 Compound filter: a(n) = P(A292583(n), A292585(n)), where P(n,k) is sequence A000027 used as a pairing function.

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