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) = A001223(n)/2 for n > 1.
a(n) = (prime(n+1) - prime(n)) / 2, where prime(n) is the n-th prime.
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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
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