A333725 Number of primes between pairs of consecutive highly composite numbers (A002182).

0, 1, 1, 2, 4, 2, 4, 2, 13, 11, 11, 20, 56, 18, 59, 58, 105, 307, 284, 278, 528, 515, 501, 241, 1684, 466, 456, 2491, 2403, 4676, 4561, 4459, 4396, 12839, 4202, 8317, 4111, 26274, 25673, 50073, 48866, 47998, 47441, 139491, 45881, 90692, 134351, 220465, 173831, 257677

Offset: 1

Author

Robin Powell, Apr 03 2020

Keywords

• nonn

Examples

			There are no primes between HCN(1) and HCN(2), so a(1) = 0. The next term a(2) is equal to 1 as 3 is the only prime between HCN(2) and HCN(3); the prime 2 is not greater than HCN(2) and so is omitted here. The first gap to contain more than one prime occurs at a(4) = 2, which alludes to 7 and 11 being the only primes contained within HCN(4) and HCN(5).
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A002182, A000720, A262501.

Programs

Mathematica

Join[{0},PrimePi[#[[2]]]-PrimePi[#[[1]]]&/@Partition[DeleteDuplicates[Table[ {n,DivisorSigma[ 0,n]},{n,2,22*10^5}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]],2,1]] (* Harvey P. Dale, Jan 09 2023 *)

Formula

a(n) = A000720(A002182(n+1)) - A000720(A002182(n)) for n > 1. - Amiram Eldar, Apr 26 2020

Extensions

More terms from Giovanni Resta, Apr 04 2020