A375245 Number of biquadratefree numbers <= n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 75, 75, 76, 77, 78, 79

Offset: 1

Author

Chai Wah Wu, Aug 07 2024

Keywords

• nonn
• easy

Comments

First differs from A309083 at n = 81: a(81) = 75, A309083(n) = 77. - Andrew Howroyd, Aug 10 2024

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A060431, A013662, A046100, A215267, A307430.

Programs

Mathematica

Accumulate[Table[Boole[Max[FactorInteger[n][[;; , 2]]] < 4], {n, 1, 100}]] (* Amiram Eldar, Aug 10 2024 *)

PARI

a(n) = sum(d=1, sqrtnint(n,4), moebius(d)*(n\d^4)) \\ Andrew Howroyd, Aug 10 2024

Python

from sympy import mobius, integer_nthroot
def A375245(n): return int(sum(mobius(k)*(n//k**4) for k in range(1, integer_nthroot(n,4)[0]+1)))

Formula

a(n) = Sum_{d>=1} mu(d)*floor(n/d^4), where mu is the Moebius function A008683.

n/a(n) converges to zeta(4).

a(n) = Sum_{k = 1..n} A307430(k).

Extensions

a(68) onwards from Andrew Howroyd, Aug 10 2024