A365210 - OEIS

%I #8 Aug 27 2023 01:46:22

%S 1,2,2,2,2,4,2,2,2,4,2,4,2,4,4,2,2,4,2,4,4,4,2,4,3,4,2,4,2,8,2,2,4,4,

%T 4,4,2,4,4,4,2,8,2,4,4,4,2,4,3,6,4,4,2,4,4,4,4,4,2,8,2,4,4,2,4,8,2,4,

%U 4,8,2,4,2,4,6,4,4,8,2,4,2,4,2,8,4,4,4

%N The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

%C First differs from A034444 at n = 25.

%C The sum of these divisors is A365211(n).

%H Amiram Eldar, <a href="/A365210/b365210.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = 2 for p = 2 and 3, and a(p^e) = e+1 for a prime p >= 5.

%F a(n) <= A000005(n), with equality if and only if n is neither divisible by 4 nor by 9.

%F a(n) >= A034444(n), with equality if and only if n is not divisible by a square of a prime >= 5.

%F a(n) = A000005(A065330(n)) * A034444(A065331(n)).

%F Dirichlet g.f.: (1-1/4^s) * (1-1/9^s) * zeta(s)^2.

%F Sum_{k=1..n} a(k) ~ (2*n/3) * (log(n) + 2*gamma - 1 + 2*log(2)/3 + log(3)/4), where gamma is Euler's constant (A001620).

%t f[p_, e_] := If[p <= 3 , 2, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, 2, f[i,2]+1));}

%Y Cf. A000005, A007310, A035218, A065330, A065331, A069733, A365211.

%Y Cf. A001620, A013661, A073002.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Aug 26 2023