A116127 Number of numbers that are congruent to {2, 4} mod 6 between prime(n) and prime(n+1) inclusive.

1, 1, 0, 2, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 2, 2, 2, 2, 2, 0, 2, 0, 2, 4, 2, 2, 0, 4, 0, 2, 2, 2, 2, 2, 0, 4, 0, 2, 0, 4, 4, 2, 0, 2, 2, 0, 4, 2, 2, 2, 0, 2, 2, 0, 4, 4, 2, 0, 2, 4, 2, 4, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 0, 4, 0, 2, 2, 2, 2, 2, 0, 2, 4, 2, 2, 2, 2, 2, 4, 0, 6, 2, 4, 2, 2, 0, 2

Offset

1,4

Comments

For n > 2,

A001223(n) = 2 iff a(n) = 0,

A001223(n) = 4 or 6 or 8 iff a(n) = 2,

A001223(n) = 10 or 12 or 14 iff a(n) = 4,

A001223(n) = 16 or 18 or 20 iff a(n) = 6,

and so on. This can be generalized to

A001223(n) = 3*k-2 or 3*k or 3*k+2 iff a(n) = k for k >= 2.

Links

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

Maple

P:= select(isprime, [seq(i, i=5..1000, 2)]):
Delta:= P[2..-1]-P[1..-2]:
f:= t -> (t + 2*(t+1 mod 3) - 2)/3:
1, 1, op(map(f, Delta)); # Robert Israel, Jun 19 2019

Mathematica

s={}; Do[c=0; Do[If[MemberQ[{2, 4}, Mod[i, 6]], c=c+1], {i, Prime[n], Prime[n+1]}]; AppendTo[s, c], {n, 105}]; s (* James C. McMahon, Aug 18 2024 *)

Prog

(Magma) [ #[ k: k in [NthPrime(n)..NthPrime(n+1)] | r eq 2 or r eq 4 where r is k mod 6 ]: n in [1..105] ]; /* Klaus Brockhaus, Apr 15 2007 */

Crossrefs

Cf. A000040 (primes), A001223 (differences between consecutive primes), A047235 (numbers congruent to {2, 4} mod 6), A002654.

Sequence in context: A029225 A341977 A309937 * A039979 A204173 A103668

Adjacent sequences: A116124 A116125 A116126 * A116128 A116129 A116130

Keyword

nonn,easy

Author

Giovanni Teofilatto, Apr 08 2007

Extensions

Edited, corrected and extended by Klaus Brockhaus, Apr 15 2007

Status

approved