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A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

%I A385007 #8 Jun 15 2025 12:53:31
%S A385007 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,1,17,18,19,20,21,22,23,24,25,26,
%T A385007 27,28,29,30,31,1,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,3,49,
%U A385007 50,51,52,53,54,55,56,57,58,59,60,61,62,63,1,65,66,67,68,69
%N A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).
%C A385007 First differs from A053165 at n = 32 = 2^5: a(32) = 1 while A053165(32) = 2.
%C A385007 First differs from A383764 at n = 32 = 2^5: a(32) = 1 while A383764(32) = 32.
%C A385007 Equivalently, a(n) is the least divisor d of n such that n/d is a 4-full number (A036967).
%H A385007 Amiram Eldar, <a href="/A385007/b385007.txt">Table of n, a(n) for n = 1..10000</a>
%F A385007 a(n) = 1 if and only if n is a 4-full number (A036967).
%F A385007 a(n) = n if and only if n is a biquadratefree number (A046100).
%F A385007 Multiplicative with a(p^e) = p^e if e <= 3, and 1 otherwise.
%F A385007 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(-s) + p^(2-2*s) - p^(1-2*s) - p^(2-3*s) + p^(3-3*s) - p^(3-4*s) + p^(-4*s)).
%F A385007 Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^6 + 1/p^8 - 1/p^9) = 0.56331392082909224894... .
%t A385007 f[p_, e_] := If[e < 4, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385007 (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] < 4, f[i, 1]^f[i, 2], 1)); }
%Y A385007 Cf. A013661, A036967, A046100, A053165, A058035, A077610, A383764.
%Y A385007 The largest unitary divisor of n that is: A000265 (odd), A006519 (power of 2), A055231 (squarefree), A057521 (powerful), A065330 (5-rough), A065331 (3-smooth), A350388 (square), A350389 (exponentially odd), A360539 (cubefree), A360540 (cubefull), A366126 (cube), A367168 (exponentially 2^n), this sequence (biquadratefree).
%K A385007 nonn,easy,mult
%O A385007 1,2
%A A385007 _Amiram Eldar_, Jun 15 2025