A080378 -id:A080378 - cp's OEIS frontend

A023520 Exponent of 2 in prime factorization of prime(n)*prime(n-1) - 1.

0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 6, 2, 3

Offset: 2

Clark Kimberling

nonn

a(n) = 1 if and only if A080378(n-1) = 2. - Robert Israel, Feb 07 2018

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A007814, A023515, A080378.

seq(padic:-ordp(ithprime(n)*ithprime(n-1)-1,2),n=2..200); # Robert Israel, Feb 07 2018

a(n) = valuation(prime(n)*prime(n-1) - 1, 2); \\ Michel Marcus, Sep 30 2013

from sympy import prime
def A023520(n): return (~(m:=prime(n)*prime(n-1)-1)& m-1).bit_length()    # Chai Wah Wu, Jul 07 2022

a(n) = A007814(A023515(n)). - Michel Marcus, Sep 30 2013

Offset set to 2 and a(2) corrected by Michel Marcus, Sep 30 2013

A330559 a(n) = (number of primes p <= prime(n) with Delta(p) == 2 (mod 4)) - (number of primes p <= prime(n) with Delta(p) == 0 (mod 4)), where Delta(p) = nextprime(p) - p.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 5, 6, 7, 6, 7, 8, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 15, 16, 15, 14, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 22, 23, 24, 25, 26, 25, 26, 25, 26, 27, 26, 27, 26, 25, 24, 25, 26, 27, 28, 29, 28

Offset: 1

N. J. A. Sloane, Dec 30 2019

nonn

Equals A330560 - A330561.
Since Delta(prime(n)) grows roughly like log n, this probably changes sign infinitely often. When is the next time a(n) is zero, or the first time a(n) < 0 (if these values exist)?
Let s = A024675, the interprimes. For each n let E(n) = number of even terms of s that are <= n, and let O(n) = number of odd terms of s that are <= n. Then a(n+1) = E(n) - O(n). That is, as we progress through s, the number of evens stays greater than the number of odds. - Clark Kimberling, Feb 26 2024

			n=6: prime(6) = 13, primes p <= 13 with Delta(p) == 2 (mod 4) are 3,5,11; primes p <= 13 with Delta(p) == 0 (mod 4) are 7,13; so a(6) = 3-2 = 1.

N. J. A. Sloane, Table of n, a(n) for n = 1..99999
StackExchange, Asymptotic Distribution of Prime Gaps in Residue Classes.

Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556, A330557, A330558, A330560, A330561.

Join[{0}, Accumulate[Mod[Differences[Prime[Range[2, 100]]], 4] - 1]] (* Paolo Xausa, Feb 05 2024 *)

A330560 a(n) = number of primes p <= prime(n) with Delta(p) == 2 (mod 4), where Delta(p) = nextprime(p) - p.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 29, 30, 30, 30, 30, 31, 31, 32, 33, 34, 35, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 43, 44, 45, 46, 47, 47, 48, 48, 49, 50, 50, 51, 51, 51, 51, 52, 53, 54

Offset: 1

N. J. A. Sloane, Dec 30 2019

nonn

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

Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556, A330557, A330558, A330559, A330561.

[#[p:p in PrimesInInterval(1,NthPrime(n))| (NextPrime(p)-p) mod 4 eq 2]:n in [1..90]]; // Marius A. Burtea, Dec 31 2019

N:= 200: # for a(1)..a(N)
P:= [seq(ithprime(i), i=1..N+1)]:
Delta:= P[2..-1]-P[1..-2] mod 4:
R:= map(charfcn[2], Delta):
ListTools:-PartialSums(R); # Robert Israel, Dec 31 2019

Accumulate[Map[Boole[Mod[#, 4] == 2]&, Differences[Prime[Range[100]]]]] (* Paolo Xausa, Feb 05 2024 *)

A259360 Initial prime in the least set of exactly n+1 consecutive primes with n gaps all multiples of 4.

Original entry on oeis.org

7, 89, 199, 883, 12401, 463, 36551, 11593, 183091, 766261, 3358169, 241603, 11739307, 9177431, 12270077, 105639091, 310523021, 297779117, 727334879, 5344989829, 1481666377, 2572421893, 1113443017, 79263248027, 84676452781

Offset: 1

Zak Seidov, Jun 24 2015

nonn

			a(6)=463 because the first set of 7 consecutive primes is {463,467,479,487,491,499,503} with 6 gaps {4,12,8,4,8,4} all multiples of 4 while the next prime after 503 is 509 and 509-503=6 is not a multiple of 4.

Giovanni Resta, Charles R Greathouse IV and Zak Seidov, Table of n, a(n) for n = 1..33

Cf. A054678, A054679, A054680, A080378, A098059.

back(p,n)=while(n,p=precprime(p-1); n--); p
v=vector(20); g=0; p=2; forprime(q=3,1e6, if((q-p)%4, if(g&&g<=#v&&v[g]==0, v[g]=back(p,g)); g=0, g++);p=q); v \\ Charles R Greathouse IV, Jul 14 2015

a(13)-a(14) corrected by Charles R Greathouse IV, Jul 14 2015

a(24)-a(25) by Zak Seidov, Jul 15 2015

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A023520 Exponent of 2 in prime factorization of prime(n)*prime(n-1) - 1.

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A330559 a(n) = (number of primes p <= prime(n) with Delta(p) == 2 (mod 4)) - (number of primes p <= prime(n) with Delta(p) == 0 (mod 4)), where Delta(p) = nextprime(p) - p.

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A330560 a(n) = number of primes p <= prime(n) with Delta(p) == 2 (mod 4), where Delta(p) = nextprime(p) - p.

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Magma

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Mathematica

A259360 Initial prime in the least set of exactly n+1 consecutive primes with n gaps all multiples of 4.

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Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions