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A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

%I A384042 #9 May 25 2025 12:36:39
%S A384042 1,1,2,2,5,2,7,4,6,5,11,4,13,7,10,8,17,6,19,10,14,11,23,8,25,13,18,14,
%T A384042 29,10,31,16,22,17,35,12,37,19,26,20,41,14,43,22,30,23,47,16,49,25,34,
%U A384042 26,53,18,55,28,38,29,59,20,61,31,42,32,65,22,67,34,46
%N A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).
%H A384042 Amiram Eldar, <a href="/A384042/b384042.txt">Table of n, a(n) for n = 1..10000</a>
%F A384042 Multiplicative with a(p^e) = (p-1)*p^(e-1) if p <= 3 and p^e if p >= 5.
%F A384042 a(n) >= A000010(n), with equality if and only if n is 3-smooth (A003586).
%F A384042 a(n) = A000010(A065331(n)) * A065330(n).
%F A384042 a(n) = 2 * n * phi(n)/phi(6*n) = n * A000010(n) / A372671(n).
%F A384042 Dirichlet g.f.: zeta(s-1) * (1-1/2^s) * (1-1/3^s).
%F A384042 Sum_{k=1..n} a(k) ~ n^2 / 3.
%t A384042 f[p_, e_] := If[p < 5, (p-1)*p^(e-1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384042 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] < 5, (f[i,1]-1)*f[i,1]^(f[i,2]-1), f[i,1]^f[i,2]));}
%Y A384042 Cf. A000010, A003586, A007310, A065330, A065331, A372671.
%Y A384042 The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), A384041 (exponentially odd), this sequence (5-rough).
%K A384042 nonn,easy,mult
%O A384042 1,3
%A A384042 _Amiram Eldar_, May 18 2025