|
%I N0030 #72 Sep 08 2022 08:44:50
%S 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 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 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 Differences between consecutive odd primes, divided by 2.
%C With an initial zero, gives the numbers of even numbers between two successive primes. - _Giovanni Teofilatto_, Nov 04 2005
%C Equal to difference between terms in A067076. - _Eric Desbiaux_, Aug 07 2010
%C 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 a(n) = 1 if, and only if, n + 1 is in A107770. - _Jason Kimberley_, Nov 13 2015
%D 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 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%H T. D. Noe, <a href="/A028334/b028334.txt">Table of n, a(n) for n = 2..10000</a>
%H 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 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 a(n) = (prime(n+1) - prime(n)) / 2, where prime(n) is the n-th prime.
%F a(n) = A047160(A024675(n-1)). - _Jason Kimberley_, Nov 12 2015
%F 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 23 - 19 = 4, so a(8) = 4/2 = 2.
%e 29 - 23 = 6, so a(9) = 6/2 = 3.
%e 31 - 29 = 2, so a(10) = 2/2 = 1.
%t Table[(Prime[n + 1] - Prime[n])/2, {n, 2, 105}] (* _Robert G. Wilson v_ *)
%t Differences[Prime[Range[2, 110]]]/2 (* _Harvey P. Dale_, Jan 25 2015 *)
%o (PARI) vector(10000,i,(prime(i+2)-prime(i+1))/2) \\ _Stanislav Sykora_, Nov 05 2014
%o (Magma) [(NthPrime(n+1)-NthPrime(n))/2: n in [2..100]]; // _Vincenzo Librandi_, Dec 12 2016
%Y Cf. A005521.
%Y Equals A001223(n)/2 for n > 1.
%Y Cf. A000230 (least prime with a gap of 2n to the next prime).
%Y Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556-A330561.
%K nonn
%O 2,3
%A _N. J. A. Sloane_
%E Replaced multiplication by division in the cross-reference _R. J. Mathar_, Jan 23 2010
%E Definition corrected by _Jonathan Sondow_, May 17 2013
%E Edited by _Franklin T. Adams-Watters_, Aug 07 2014
|
