A380254 Number of powerful numbers (in A001694) that do not exceed primorial A002110(n).
1, 1, 2, 7, 22, 85, 330, 1433, 6450, 31555, 172023, 964560, 5891154, 37807505, 248226019, 1702890101, 12401685616, 95277158949, 744210074157, 6091922351106, 51332717836692, 438592279944173, 3898316990125822, 35515462315592564, 335052677538616216, 3299888425002527366
Offset: 0
Examples
Let P = A002110 and let s = A001694.
a(0) = 1 since P(0) = 1, and the set s(1) = {1} contains k that do not exceed 1.
a(1) = 1 since P(1) = 2, and the set s(1) = {1} contains k <= 2.
a(2) = 2 since P(2) = 6, and the set s(1..2) = {1, 4} contains k <= 6.
a(3) = 7 since P(3) = 30, and the set s(1..7) = {1, 4, 8, 9, 16, 25, 27} contains k <= 30.
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.
Links
- Chai Wah Wu, Table of n, a(n) for n = 0..26
Programs
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Mathematica
f[x_] := Sum[If[SquareFreeQ[ii], Floor[Sqrt[x/ii^3]], 0], {ii, x^(1/3)}]; Table[f[#[[k + 1]]], {k, 0, Length[#] - 1}] &[ FoldList[Times, 1, Prime[Range[12] ] ] ] (* function f after Robert G. Wilson v at A118896 *) -
Python
from math import isqrt from sympy import primorial, integer_nthroot, mobius def A380254(n): def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1))) if n == 0: return 1 m = primorial(n) c, l, j = squarefreepi(integer_nthroot(m, 3)[0]), 0, isqrt(m) while j>1: k2 = integer_nthroot(m//j**2,3)[0]+1 w = squarefreepi(k2-1) c += j*(w-l) l, j = w, isqrt(m//k2**3) return c-l # Chai Wah Wu, Jan 24 2025
Extensions
a(18)-a(25) from Chai Wah Wu, Jan 24 2025
Comments