cp's OEIS Frontend

A340041 The prime gap, divided by two, which surrounds p#.

%I A340041 #19 Mar 07 2021 00:11:01
%S A340041 1,1,6,1,9,24,23,40,51,37,60,36,68,87,66,84,99,95,115,88,117,143,51,
%T A340041 177,182,168,139,243,221,193,204,516,260,154,182,306,239,216,191,211,
%U A340041 303,263,672,303,615,417,312,378,275,375,322,445,312,294,354,492,399,348,461
%N A340041 The prime gap, divided by two, which surrounds p#.
%C A340041 If p and q are consecutive primes, we say here that there is a gap of q-p. (Other sequences use different definitions of "gap".) - _N. J. A. Sloane_, Mar 07 2021
%C A340041 Records: 1, 6, 9, 24, 40, 51, 60, 68, 87, 99, 115, 117, 143, 177, 182, 243, 516, 672, 855, 915, 925, 1100, 1139, 1620, 1863, 2272, 2842, 4177, 4190, 5025, 5692, 6254, 6413, 6879, 7914, 8026, 9928, 10604, ..., .
%F A340041 a(n) = (A006862(n) - A007014(n))/2 = (A038711(n) + A060270(n))/2.
%F A340041 a(n) =  A058044(n)/2. - _Hugo Pfoertner_, Jan 22 2021
%e A340041 For a(1), there are two contiguous primes {2, 3} with 2 being 2#. The prime gap is 1. However, the two primes do not surround 2#, so a(1) like A340013(2) is undefined.
%e A340041 For a(2), the prime gap contains {5, 6, 7}, with 3# = 6 in the middle. The prime gap is 2, therefore a(2) = 1;
%e A340041 For a(3), the prime gap contains {29, 30, 31}, with 5# = 30  in the middle. The prime gap is 2, therefore a(3) = 1.
%e A340041 For a(4), the prime gap contains {199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211}, with 7# =  205 in the middle. The prime gap is 12, therefore a(4) = 6. etc.
%t A340041 a[n_] := Block[{p = Times @@ Prime@ Range@ n}, (NextPrime[p, 1] - NextPrime[p, -1])/2]; a[1] = 0; Array[a, 60]
%Y A340041 Cf. A006862, A007014, A038711, A060270, A340013 (analog for n!).
%K A340041 nonn
%O A340041 2,3
%A A340041 _Robert G. Wilson v_, Jan 22 2021