A028334 Differences between consecutive odd primes, divided by 2. (Formerly N0030)

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

2,3

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

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.

Formula

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

Example

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.

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 *)

Prog

(PARI) vector(10000, i, (prime(i+2)-prime(i+1))/2) \\ Stanislav Sykora, Nov 05 2014
(Magma) [(NthPrime(n+1)-NthPrime(n))/2: n in [2..100]]; // Vincenzo Librandi, Dec 12 2016

Crossrefs

Cf. A005521.

Equals A001223(n)/2 for n > 1.

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.

Sequence in context: A361477 A331366 A274225 * A083269 A097306 A102632

Adjacent sequences: A028331 A028332 A028333 * A028335 A028336 A028337

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved