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A384057 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a 3-smooth number.

%I A384057 #11 May 21 2025 01:32:56
%S A384057 1,2,3,4,4,6,6,8,9,8,10,12,12,12,12,16,16,18,18,16,18,20,22,24,24,24,
%T A384057 27,24,28,24,30,32,30,32,24,36,36,36,36,32,40,36,42,40,36,44,46,48,48,
%U A384057 48,48,48,52,54,40,48,54,56,58,48,60,60,54,64,48,60,66,64
%N A384057 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a 3-smooth number.
%C A384057 First differs A372671 from at n = 25.
%H A384057 Amiram Eldar, <a href="/A384057/b384057.txt">Table of n, a(n) for n = 1..10000</a>
%F A384057 Multiplicative with a(p^e) = p^e if p <= 3, and p^e-1 if p >= 5.
%F A384057 a(n) = n * A047994(n) / A384058(n).
%F A384057 a(n) = A047994(A065330(n)) * A065331(n).
%F A384057 Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1-1/2^s)/(1-1/2^(s-1)+1/2^(2*s-1))) * ((1-1/3^s)/(1-2/3^s+1/3^(2*s-1))) * Product_{p prime} (1 - 2/p^s + 1/p^(2*s-1)).
%F A384057 Sum_{k=1..n} a(k) ~ (36/55) * c * n^2, where c = Product_{p prime} (1 - 1/(p*(p+1))) = A065463.
%F A384057 In general, the average order of the number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a p-smooth number (i.e., not divisible by any prime larger than the prime p) is (1/2) * Product_{q prime <= p} (1 + 1/(q^2+q-1)) * A065463 * n^2.
%t A384057 f[p_, e_] := p^e - If[p < 5, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384057 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,1] < 5, 0, 1));}
%Y A384057 Unitary analog of A372671.
%Y A384057 Cf. A003586, A065330, A065331, A065463, A372671, A384046, A384047.
%Y A384057 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), this sequence (3-smooth), A384058 (5-rough).
%K A384057 nonn,easy,mult
%O A384057 1,2
%A A384057 _Amiram Eldar_, May 18 2025