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Number of biquadratefree numbers <= n.
3

%I #25 Aug 10 2024 21:39:14

%S 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,

%T 26,27,28,29,30,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,45,46,

%U 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

%N Number of biquadratefree numbers <= n.

%C First differs from A309083 at n = 81: a(81) = 75, A309083(n) = 77. - _Andrew Howroyd_, Aug 10 2024

%H Andrew Howroyd, <a href="/A375245/b375245.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%t Accumulate[Table[Boole[Max[FactorInteger[n][[;; , 2]]] < 4], {n, 1, 100}]] (* _Amiram Eldar_, Aug 10 2024 *)

%o (Python)

%o from sympy import mobius, integer_nthroot

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

%o (PARI) a(n) = sum(d=1, sqrtnint(n,4), moebius(d)*(n\d^4)) \\ _Andrew Howroyd_, Aug 10 2024

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

%K nonn,easy

%O 1,2

%A _Chai Wah Wu_, Aug 07 2024

%E a(68) onwards from _Andrew Howroyd_, Aug 10 2024