A287484 Number of squarefree k with A002110(n) <= k < A002110(n+1) such that A001221(k) = n.

1, 3, 7, 19, 58, 152, 422, 995, 2359, 6294, 14507, 36370, 88198, 187786, 386993, 840033, 1901930, 3851372, 8088478, 16388857, 30001902, 56613547, 103229263, 193020113, 389750880, 759988983, 1359250012, 2350842201, 3737393021, 5748044055, 10843131073, 19774152370

Offset

0,2

Comments

Primorial A002110(n) is the smallest squarefree number with n prime factors. a(n) is a list of squarefree numbers with n prime factors greater than and including A002110(n) but less than A002110(n+1).

a(1) counts the first primes less than 6.

a(2) counts the first squarefree semiprimes (A006881) less than 30.

a(3) counts the smallest terms of A033992 less than 210, etc.

Links

Table of n, a(n) for n=0..31.

Eric Weisstein's World of Mathematics, Primorial

Eric Weisstein's World of Mathematics, Squarefree

Example

Let p_n# = A002110(n).
a(0) = 1 since the only squarefree number between p_0# and (p_1# - 1) (i.e., 1 and 1) with 0 prime factors is 1.
a(1) = 3 since for p_1# <= k <= (p_2# - 1), i.e., 2 <= k <= 5, there are three primes {2, 3, 5}.
a(2) = 7 since we find the squarefree semiprimes {6, 10, 14, 15, 21, 22, 26} between 6 and 29 inclusive.

Mathematica

Table[Count[Range[#, Prime[n + 1] # - 1] &@ Product[Prime@ i, {i, n}], k_ /; And[SquareFreeQ@ k, PrimeOmega@ k == n]], {n, 0, 6}]

Crossrefs

Cf. A001221, A002110, A005117, A006881, A033992.

Sequence in context: A263334 A005506 A268125 * A201169 A148667 A148668

Adjacent sequences: A287481 A287482 A287483 * A287485 A287486 A287487

Keyword

nonn,hard

Author

Michael De Vlieger, May 25 2017

Extensions

a(25)-a(31) from David A. Corneth, May 31 2017

Status

approved