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A384058 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 5-rough number (A007310).

%I A384058 #10 May 21 2025 01:33:02
%S A384058 1,1,2,3,5,2,7,7,8,5,11,6,13,7,10,15,17,8,19,15,14,11,23,14,25,13,26,
%T A384058 21,29,10,31,31,22,17,35,24,37,19,26,35,41,14,43,33,40,23,47,30,49,25,
%U A384058 34,39,53,26,55,49,38,29,59,30,61,31,56,63,65,22,67,51,46
%N A384058 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 5-rough number (A007310).
%H A384058 Amiram Eldar, <a href="/A384058/b384058.txt">Table of n, a(n) for n = 1..10000</a>
%F A384058 Multiplicative with a(p^e) = p^e-1 if p <= 3, and p^e if p >= 5.
%F A384058 a(n) = n * A047994(n) / A384057(n).
%F A384058 a(n) = A047994(A065331(n)) * A065330(n).
%F A384058 Dirichlet g.f.: zeta(s-1) * ((1 - 1/2^(s-1) + 1/2^(2*s-1))/(1 - 1/2^s)) * ((1 - 2/3^s + 1/3^(2*s-1))/(1 - 1/3^s)).
%F A384058 Sum_{k=1..n} a(k) ~ (55/144) * n^2.
%F A384058 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-rough number (i.e., not divisible by any prime smaller than the prime p) is (1/2) * Product_{q prime <= p} (1 - 1/q + 1/(q+1)) * n^2.
%t A384058 f[p_, e_] := p^e - If[p < 5, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384058 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,1] < 5, 1, 0));}
%Y A384058 Unitary analog of A384042.
%Y A384058 Cf. A007310, A065330, A065331, A384046, A384047.
%Y A384058 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), A384057 (3-smooth), this sequence (5-rough).
%K A384058 nonn,easy,mult
%O A384058 1,3
%A A384058 _Amiram Eldar_, May 18 2025