A380254 -id:A380254 - cp's OEIS frontend

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

Michael De Vlieger, Jan 21 2025

nonn

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

			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.

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

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

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

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

A380430 Number of powerful numbers k that are not powers of primes (i.e., k is in A286708) that do not exceed the primorial number A002110(n).

Original entry on oeis.org

0, 0, 0, 0, 7, 50, 254, 1245, 5898, 29600, 163705, 925977, 5690175, 36681963, 241663896, 1662446097, 12134853382, 93406989325, 730785520398, 5990426525483, 50538885715526, 432266550168097, 3845700235189327, 35065304557027821, 334652745159828239, 3262707438761612651

Offset: 0

Michael De Vlieger, Jan 24 2025

			Let P = A002110 and let s = A286708 = A001694 \ A246547 \ {1}.
a(0..3) = 0 since the smallest number in s is 36.
a(4) = 7 since P(4) = 210 and numbers in s that are less than 210 include {36, 72, 100, 108, 144, 196, 200}, etc.

Cf. A001694, A126706, A246547, A286708, A380254, A380402.

Table[-1 + Sum[If[MoebiusMu[j] != 0, Floor[Sqrt[#/j^3]], 0], {j, #^(1/3)}] - Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor[Log2[#]]}] &[Product[Prime[i], {i, n}]], {n, 0, 12} ]

from math import isqrt
from sympy import primorial, primepi, integer_nthroot, mobius
def A380430(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    if n == 0: return 0
    m = primorial(n)
    c, l, j = int(squarefreepi(integer_nthroot(m, 3)[0])-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))-1), 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, Feb 25 2025

a(n) = A380254(n) - A380402(n) - 1.

a(n) <= A380403(n) since A286708 is a proper subset of A126706.

cp's OEIS Frontend

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

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