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

Author

Giovanni Teofilatto, Apr 08 2007

Keywords

• nonn
• easy

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

Crossrefs

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

Programs

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 */

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 *)

Extensions

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