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A380254 Number of powerful numbers (in A001694) that do not exceed primorial A002110(n).

%I A380254 #24 Jan 25 2025 02:43:29
%S A380254 1,1,2,7,22,85,330,1433,6450,31555,172023,964560,5891154,37807505,
%T A380254 248226019,1702890101,12401685616,95277158949,744210074157,
%U A380254 6091922351106,51332717836692,438592279944173,3898316990125822,35515462315592564,335052677538616216,3299888425002527366
%N A380254 Number of powerful numbers (in A001694) that do not exceed primorial A002110(n).
%C A380254 In other words, A001694(a(n)) is the largest powerful number less than or equal to A002110(n).
%H A380254 Chai Wah Wu, <a href="/A380254/b380254.txt">Table of n, a(n) for n = 0..26</a>
%e A380254 Let P = A002110 and let s = A001694.
%e A380254 a(0) = 1 since P(0) = 1, and the set s(1) = {1} contains k that do not exceed 1.
%e A380254 a(1) = 1 since P(1) = 2, and the set s(1) = {1} contains k <= 2.
%e A380254 a(2) = 2 since P(2) = 6, and the set s(1..2) = {1, 4} contains k <= 6.
%e A380254 a(3) = 7 since P(3) = 30, and the set s(1..7) = {1, 4, 8, 9, 16, 25, 27} contains k <= 30.
%e A380254 a(4) = 22 since P(4) = 210, and the set s(1..19) = {1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200} contains k <= 210, etc.
%t A380254 f[x_] := Sum[If[SquareFreeQ[ii], Floor[Sqrt[x/ii^3]], 0], {ii, x^(1/3)}];
%t A380254 Table[f[#[[k + 1]]], {k, 0, Length[#] - 1}] &[
%t A380254   FoldList[Times, 1, Prime[Range[12] ] ] ] (* function f after _Robert G. Wilson v_ at A118896 *)
%o A380254 (Python)
%o A380254 from math import isqrt
%o A380254 from sympy import primorial, integer_nthroot, mobius
%o A380254 def A380254(n):
%o A380254     def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o A380254     if n == 0: return 1
%o A380254     m = primorial(n)
%o A380254     c, l, j = squarefreepi(integer_nthroot(m, 3)[0]), 0, isqrt(m)
%o A380254     while j>1:
%o A380254         k2 = integer_nthroot(m//j**2,3)[0]+1
%o A380254         w = squarefreepi(k2-1)
%o A380254         c += j*(w-l)
%o A380254         l, j = w, isqrt(m//k2**3)
%o A380254     return c-l # _Chai Wah Wu_, Jan 24 2025
%Y A380254 Cf. A001694, A002110, A062762, A118896, A380337.
%K A380254 nonn,hard
%O A380254 0,3
%A A380254 _Michael De Vlieger_, Jan 19 2025
%E A380254 a(18)-a(25) from _Chai Wah Wu_, Jan 24 2025