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A380403
Number of integers k that are neither squarefree nor prime powers (in A126706) and that do not exceed primorial A002110(n).
0
0, 0, 0, 5, 67, 871, 11693, 199976, 3802411, 87466676, 2536583089, 78634293907, 2909470106300, 119288281458176, 5129396144497507, 241081619059363357, 12777325812023481231, 753862222923258499554
OFFSET
0,4
FORMULA
a(n) = P(n) - (Sum_{k=2..floor(log_2(P(n)))}, pi(floor(P(n)^(1/k)))) - Sum_{k=1..floor(sqrt(P(n)))}, mu(k)*floor(P(n)/(k^2)), where P(n) = A002110(n).
a(n) = A002110(n) - A380402(n) - A158341(n) - 1.
EXAMPLE
Let s = A126706 and let P(n) = A002110(n).
a(0..2) = 0 since P(0..2) = {1, 2, 6}, and the smallest number in s is 12.
a(3) = 5 since P(3) = 30, and the set s(1..6) = {12, 18, 20, 24, 28} contains k <= 30.
a(4) = 67 since P(4) = 210, and the set s(1..67) = {12, 18, 20, ..., 207, 208} contains k <= 210, etc.
MATHEMATICA
Table[# - Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor[Log2[#]]}] - Sum[MoebiusMu[k]*Floor[#/(k^2)], {k, Floor[Sqrt[#]]}] &[Product[Prime[i], {i, n}]], {n, 0, 12}]
PROG
(Python)
from math import isqrt
from sympy import primorial, primepi, integer_nthroot, mobius
def A380403(n):
if n == 0: return 0
m = primorial(n)
return int(-sum(primepi(integer_nthroot(m, k)[0]) for k in range(2, m.bit_length()))-sum(mobius(k)*(m//k**2) for k in range(2, isqrt(m)+1))) # Chai Wah Wu, Jan 24 2025
CROSSREFS
KEYWORD
nonn,hard,more,new
AUTHOR
Michael De Vlieger, Jan 23 2025
EXTENSIONS
Offset changed to 0 by Jinyuan Wang, Jan 24 2025
a(16) from Chai Wah Wu, Jan 24 2025
a(17) from Chai Wah Wu, Jan 25 2025
STATUS
approved