A368336 - OEIS

%I #7 Dec 21 2023 21:16:06

%S 1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,5,1,1,1,3,1,1,1,3,1,1,4,3,1,1,1,5,1,1,

%T 1,3,1,1,1,3,1,1,1,3,1,1,1,5,1,1,1,3,1,4,1,3,1,1,1,3,1,1,1,7,1,1,1,3,

%U 1,1,1,3,1,1,1,3,1,1,1,5,4,1,1,3,1,1,1

%N The number of divisors of the largest term of A072873 that divides of n.

%H Amiram Eldar, <a href="/A368336/b368336.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000005(A327939(n)).

%F Multiplicative with a(p^e) = e - (e mod p) + 1.

%F a(n) >= 1, with equality if and only if n is in A048103.

%F a(n) <= A000005(n), with equality if and only if n is in A072873.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + p/(p^p-1)) = 1.86196549645040699446... .

%t f[p_, e_] := (e - Mod[e, 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~, f[i,2] - f[i,2]%f[i,1] + 1);}

%Y Cf. A000005, A048103, A072873, A327939, A365632, A368331, A368335.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Dec 21 2023