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A339385 a(n) = (smallest prime >= A002182(n)) - (largest prime <= A002182(n)).

%I A339385 #19 Dec 25 2020 10:45:13
%S A339385 0,2,2,2,6,6,6,2,14,2,2,8,8,14,18,24,18,12,2,12,14,12,30,32,18,24,2,
%T A339385 40,2,30,26,30,18,14,34,14,40,18,20,40,34,36,18,20,42,120,90,24,34,52,
%U A339385 44,72,20,20,38,44,42,54,24,60,72,20,72,30,20,20,24,70
%N A339385 a(n) = (smallest prime >= A002182(n)) - (largest prime <= A002182(n)).
%C A339385 The prime gap size at the n-th highly composite number A002182(n), for n > 2.
%C A339385 The obtained arithmetic mean of the normalized gap size, i.e., a(n)/log(A002182(n)), for the terms 3..10000 is 3.030.
%C A339385 From Gauss's prime counting function approximation, the expected gap size should be approximately log(A002182), however, the observed values seem to be closer to log(A002182(n)^3).
%C A339385 The maximum merit (= a(n)/log(prevprime(A002182))) in the range 3..10000 is 12.96 and is obtained for n = 6911.
%H A339385 A.H.M. Smeets, <a href="/A339385/b339385.txt">Table of n, a(n) for n = 2..10000</a>
%F A339385 a(n) = A324385(n)+A141345(n), for n > 1.
%t A339385 s = {}; dm = 1; Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[s, NextPrime[n - 1] - NextPrime[n + 1, -1]]], {n, 2, 10^6}]; s (* _Amiram Eldar_, Dec 02 2020 *)
%t A339385 {0}~Join~Map[Subtract @@ NextPrime[#, {1, -1}] &, Import["https://oeis.org/A002182/b002182.txt", "Data"][[3 ;; 10^3, -1]] ] (* _Michael De Vlieger_, Dec 10 2020 *)
%o A339385 (PARI) lista(nn) = my(r=1); forstep(n=2, nn, 2, if(numdiv(n)>r, r=numdiv(n); print1(nextprime(n) - precprime(n), ", "))); \\ _Michel Marcus_, Dec 03 2020
%Y A339385 Cf. A005250, A141345, A324385.
%K A339385 nonn
%O A339385 2,2
%A A339385 _A.H.M. Smeets_, Dec 02 2020