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A028334 Differences between consecutive odd primes, divided by 2.

%I A028334 N0030 #76 Jul 17 2024 09:43:19
%S A028334 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,
%T A028334 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,
%U A028334 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
%N A028334 Differences between consecutive odd primes, divided by 2.
%C A028334 With an initial zero, gives the numbers of even numbers between two successive primes. - _Giovanni Teofilatto_, Nov 04 2005
%C A028334 Equal to difference between terms in A067076. - _Eric Desbiaux_, Aug 07 2010
%C A028334 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
%C A028334 a(n) = 1 if, and only if, n + 1 is in A107770. - _Jason Kimberley_, Nov 13 2015
%D A028334 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.
%D A028334 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%H A028334 T. D. Noe, <a href="/A028334/b028334.txt">Table of n, a(n) for n = 2..10000</a>
%H A028334 Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A028334 Yitang Zhang, <a href="http://dx.doi.org/10.4007/annals.2014.179.3.7">Bounded gaps between primes</a>, Annals of Mathematics, Pages 1121-1174 from Volume 179 (2014), Issue 3.
%F A028334 a(n) = A001223(n)/2 for n > 1.
%F A028334 a(n) = (prime(n+1) - prime(n)) / 2, where prime(n) is the n-th prime.
%F A028334 a(n) = A047160(A024675(n-1)). - _Jason Kimberley_, Nov 12 2015
%F A028334 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
%e A028334 23 - 19 = 4, so a(8) = 4/2 = 2.
%e A028334 29 - 23 = 6, so a(9) = 6/2 = 3.
%e A028334 31 - 29 = 2, so a(10) = 2/2 = 1.
%t A028334 Table[(Prime[n + 1] - Prime[n])/2, {n, 2, 105}] (* _Robert G. Wilson v_ *)
%t A028334 Differences[Prime[Range[2, 110]]]/2 (* _Harvey P. Dale_, Jan 25 2015 *)
%o A028334 (PARI) vector(10000,i,(prime(i+2)-prime(i+1))/2) \\ _Stanislav Sykora_, Nov 05 2014
%o A028334 (Magma) [(NthPrime(n+1)-NthPrime(n))/2: n in [2..100]]; // _Vincenzo Librandi_, Dec 12 2016
%o A028334 (SageMath)
%o A028334 def A028334(n): return (nth_prime(n+1) - nth_prime(n))//2
%o A028334 [A028334(n) for n in range(2,101)] # _G. C. Greubel_, Jul 17 2024
%Y A028334 Cf. A005521.
%Y A028334 Cf. A000230 (least prime with a gap of 2n to the next prime).
%Y A028334 Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556 - A330561.
%K A028334 nonn
%O A028334 2,3
%A A028334 _N. J. A. Sloane_
%E A028334 Replaced multiplication by division in the cross-reference _R. J. Mathar_, Jan 23 2010
%E A028334 Definition corrected by _Jonathan Sondow_, May 17 2013
%E A028334 Edited by _Franklin T. Adams-Watters_, Aug 07 2014