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

1,4

Links

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

Formula

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

Example

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).

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 *)

Crossrefs

Cf. A002182, A000720, A262501.

Sequence in context: A334970 A274708 A280638 * A163894 A307095 A032059

Adjacent sequences: A333722 A333723 A333724 * A333726 A333727 A333728

Keyword

nonn

Author

Robin Powell, Apr 03 2020

Extensions

More terms from Giovanni Resta, Apr 04 2020

Status

approved