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

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, 40, 2, 30, 26, 30, 18, 14, 34, 14, 40, 18, 20, 40, 34, 36, 18, 20, 42, 120, 90, 24, 34, 52, 44, 72, 20, 20, 38, 44, 42, 54, 24, 60, 72, 20, 72, 30, 20, 20, 24, 70

Offset: 2

Author

A.H.M. Smeets, Dec 02 2020

Keywords

• nonn

Comments

The prime gap size at the n-th highly composite number A002182(n), for n > 2.

The obtained arithmetic mean of the normalized gap size, i.e., a(n)/log(A002182(n)), for the terms 3..10000 is 3.030.

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).

The maximum merit (= a(n)/log(prevprime(A002182))) in the range 3..10000 is 12.96 and is obtained for n = 6911.

Links

• A.H.M. Smeets, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A005250, A141345, A324385.

Programs

Mathematica

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 *)
{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 *)

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

Formula

a(n) = A324385(n)+A141345(n), for n > 1.