A380337 Number of perfect powers (in A001597) that do not exceed primorial A002110(n).

1, 1, 2, 7, 19, 63, 208, 802, 3344, 15576, 82368, 453834, 2743903, 17510668, 114616907, 785002449, 5711892439, 43861741799, 342522899289, 2803468693325, 23621594605383, 201819398349092, 1793794228847381, 16342173067958793, 154171432351500060, 1518411003599957803

Offset: 0

Author

Michael De Vlieger, Jan 21 2025

Keywords

• nonn

Comments

In other words, A001597(a(n)) is the largest perfect power less than or equal to A002110(n).

Examples

			Let P = A002110 and let s = A001597.
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) = 19 since P(4) = 210, and the set s(1..19) = {1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196} contains k <= 210, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..632

Crossrefs

Cf. A001597, A002110, A070228, A070428, A380254.

Programs

Mathematica

Map[1 - Sum[MoebiusMu[k]*Floor[#^(1/k) - 1], {k, 2, Floor[Log2[#]]}] &, FoldList[Times, 1, Prime[Range[30]]] ]

Python

from sympy import primorial, mobius, integer_nthroot
def A380337(n):
    if n == 0: return 1
    p = primorial(n)
    return int(1-sum(mobius(k)*(integer_nthroot(p,k)[0]-1) for k in range(2,p.bit_length()))) # Chai Wah Wu, Jan 23 2025