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

Author

Michael De Vlieger, May 25 2017

Keywords

• nonn
• hard

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.

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Primorial
• Eric Weisstein's World of Mathematics, Squarefree

Crossrefs

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

Programs

Mathematica

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

Extensions

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