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Number of squarefree k with A002110(n) <= k < A002110(n+1) such that A001221(k) = n.
3

%I #15 May 31 2017 15:57:08

%S 1,3,7,19,58,152,422,995,2359,6294,14507,36370,88198,187786,386993,

%T 840033,1901930,3851372,8088478,16388857,30001902,56613547,103229263,

%U 193020113,389750880,759988983,1359250012,2350842201,3737393021,5748044055,10843131073,19774152370

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

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

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

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Primorial.html">Primorial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>

%e Let p_n# = A002110(n).

%e 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 a(1) = 3 since for p_1# <= k <= (p_2# - 1), i.e., 2 <= k <= 5, there are three primes {2, 3, 5}.

%e a(2) = 7 since we find the squarefree semiprimes {6, 10, 14, 15, 21, 22, 26} between 6 and 29 inclusive.

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

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

%K nonn,hard

%O 0,2

%A _Michael De Vlieger_, May 25 2017

%E a(25)-a(31) from _David A. Corneth_, May 31 2017