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

%I A287484 #16 Feb 16 2025 08:33:46
%S A287484 1,3,7,19,58,152,422,995,2359,6294,14507,36370,88198,187786,386993,
%T A287484 840033,1901930,3851372,8088478,16388857,30001902,56613547,103229263,
%U A287484 193020113,389750880,759988983,1359250012,2350842201,3737393021,5748044055,10843131073,19774152370
%N A287484 Number of squarefree k with A002110(n) <= k < A002110(n+1) such that A001221(k) = n.
%C A287484 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).
%C A287484 a(1) counts the first primes less than 6.
%C A287484 a(2) counts the first squarefree semiprimes (A006881) less than 30.
%C A287484 a(3) counts the smallest terms of A033992 less than 210, etc.
%H A287484 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Primorial.html">Primorial</a>
%H A287484 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%e A287484 Let p_n# = A002110(n).
%e A287484 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.
%e A287484 a(1) = 3 since for p_1# <= k <= (p_2# - 1), i.e., 2 <= k <= 5, there are three primes {2, 3, 5}.
%e A287484 a(2) = 7 since we find the squarefree semiprimes {6, 10, 14, 15, 21, 22, 26} between 6 and 29 inclusive.
%t A287484 Table[Count[Range[#, Prime[n + 1] # - 1] &@ Product[Prime@ i, {i, n}], k_ /; And[SquareFreeQ@ k, PrimeOmega@ k == n]], {n, 0, 6}]
%Y A287484 Cf. A001221, A002110, A005117, A006881, A033992.
%K A287484 nonn,hard
%O A287484 0,2
%A A287484 _Michael De Vlieger_, May 25 2017
%E A287484 a(25)-a(31) from _David A. Corneth_, May 31 2017