A368328 - OEIS

%I #8 Dec 21 2023 21:14:56

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

%T 1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,5,1,1,1,1,

%U 1,1,1,2,1,1,1,1,1,1,1,3,2,1,1,1,1,1,1

%N The number of terms of A054743 that divide n.

%C The number of divisors d of n such that e > p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).

%C The largest of these divisors is A368329(n).

%H Amiram Eldar, <a href="/A368328/b368328.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = e - p + 1 if e > p.

%F a(n) >= 1, with equality if and only if n is in A207481.

%F Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^((p+1)*s)).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^p)) = 1.27325025767774256043... .

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

%Y Cf. A054743, A207481, A365632, A368329, A368330, A368331, A368332.

%K nonn,easy,mult

%O 1,8

%A _Amiram Eldar_, Dec 21 2023