A028334 Differences between consecutive odd primes, divided by 2.

1, 1, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 3, 1, 3, 2, 1, 3, 2, 3, 4, 2, 1, 2, 1, 2, 7, 2, 3, 1, 5, 1, 3, 3, 2, 3, 3, 1, 5, 1, 2, 1, 6, 6, 2, 1, 2, 3, 1, 5, 3, 3, 3, 1, 3, 2, 1, 5, 7, 2, 1, 2, 7, 3, 5, 1, 2, 3, 4, 3, 3, 2, 3, 4, 2, 4, 5, 1, 5, 1, 3, 2, 3, 4, 2, 1, 2, 6, 4, 2, 4, 2, 3, 6, 1, 9, 3, 5, 3, 3, 1, 3

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

Comments

With an initial zero, gives the numbers of even numbers between two successive primes. - Giovanni Teofilatto, Nov 04 2005

Equal to difference between terms in A067076. - Eric Desbiaux, Aug 07 2010

The twin prime conjecture is that a(n) = 1 infinitely often. Yitang Zhang has proved that a(n) < 3.5 x 10^7 infinitely often. - Jonathan Sondow, May 17 2013

a(n) = 1 if, and only if, n + 1 is in A107770. - Jason Kimberley, Nov 13 2015

Examples

			23 - 19 = 4, so a(8) = 4/2 = 2.
29 - 23 = 6, so a(9) = 6/2 = 3.
31 - 29 = 2, so a(10) = 2/2 = 1.
		

References

• Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 2..10000
• Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Yitang Zhang, Bounded gaps between primes, Annals of Mathematics, Pages 1121-1174 from Volume 179 (2014), Issue 3.

Crossrefs

Cf. A005521.

Cf. A000230 (least prime with a gap of 2n to the next prime).

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

Programs

Magma

[(NthPrime(n+1)-NthPrime(n))/2: n in [2..100]]; // Vincenzo Librandi, Dec 12 2016

Mathematica

Table[(Prime[n + 1] - Prime[n])/2, {n, 2, 105}] (* Robert G. Wilson v *)
Differences[Prime[Range[2, 110]]]/2 (* Harvey P. Dale, Jan 25 2015 *)

PARI

vector(10000,i,(prime(i+2)-prime(i+1))/2) \\ Stanislav Sykora, Nov 05 2014

SageMath

def A028334(n): return (nth_prime(n+1) - nth_prime(n))//2
[A028334(n) for n in range(2,101)] # G. C. Greubel, Jul 17 2024

Formula

a(n) = A001223(n)/2 for n > 1.

a(n) = (prime(n+1) - prime(n)) / 2, where prime(n) is the n-th prime.

a(n) = A047160(A024675(n-1)). - Jason Kimberley, Nov 12 2015

G.f.: (b(x)/((x + 1)/((1 - x)) - 1) - 1 - x/2)/x, where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 10 2016

Extensions

Replaced multiplication by division in the cross-reference R. J. Mathar, Jan 23 2010

Definition corrected by Jonathan Sondow, May 17 2013

Edited by Franklin T. Adams-Watters, Aug 07 2014