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A333725
Number of primes between pairs of consecutive highly composite numbers (A002182).
1
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
KEYWORD
nonn
AUTHOR
Robin Powell, Apr 03 2020
EXTENSIONS
More terms from Giovanni Resta, Apr 04 2020
STATUS
approved