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

0,3

Comments

In other words, A001694(a(n)) is the largest powerful number less than or equal to A002110(n).

Links

Chai Wah Wu, Table of n, a(n) for n = 0..26

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A001694, A002110, A062762, A118896, A380337.

Sequence in context: A014558 A150322 A150323 * A150324 A150325 A102311

Adjacent sequences: A380251 A380252 A380253 * A380255 A380256 A380257

Keyword

nonn,hard

Author

Michael De Vlieger, Jan 19 2025

Extensions

a(18)-a(25) from Chai Wah Wu, Jan 24 2025

Status

approved