A365211 - OEIS

%I #7 Aug 27 2023 01:45:52

%S 1,3,4,5,6,12,8,9,10,18,12,20,14,24,24,17,18,30,20,30,32,36,24,36,31,

%T 42,28,40,30,72,32,33,48,54,48,50,38,60,56,54,42,96,44,60,60,72,48,68,

%U 57,93,72,70,54,84,72,72,80,90,60,120,62,96,80,65,84,144,68

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

%C First differs from A034448 at n = 25.

%C The number of these divisors is A365210(n).

%H Amiram Eldar, <a href="/A365211/b365211.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) <= A000203(n), with equality if and only if n is neither divisible by 4 nor by 9.

%F a(n) >= A034448(n), with equality if and only if n is not divisible by a square of a prime >= 5.

%F a(n) = A000203(A065330(n)) * A034448(A065331(n)).

%F Dirichlet g.f.: (1 - 1/2^(2*s-1)) * (1 - 1/3^(2*s-1)) * zeta(s)*zeta(s-1).

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 91*Pi^2/1296 = 0.69300463... .

%t f[p_, e_] := If[p <= 3 , 1 + p^e, (p^(e+1)-1)/(p-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, 1 + f[i,1]^f[i,2], (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1)));}

%Y Cf. A000203, A007310, A034448, A065330, A065331, A069184, A186099, A365210.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Aug 26 2023