A366309 - OEIS

%I #8 Oct 07 2023 08:47:42

%S 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,1,1,1,

%T 1,4,1,1,1,2,1,1,1,2,2,1,1,1,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,2,1,1,1,2,

%U 1,1,1,4,1,1,2,2,1,1,1,1,1,1,1,2,1,1,1

%N The number of infinitary divisors of n that are terms of A366243.

%H Amiram Eldar, <a href="/A366309/b366309.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = 2^A139352(e).

%F a(n) = 2^A366247(n).

%F a(n) = A037445(n)/A366308(n).

%F a(n) = A037445(A366245(n)).

%F a(n) >= 1, with equality if and only if n is in A366242.

%F a(n) <= A037445(n), with equality if and only if n is in A366243.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/p)*(1 + Sum_{k>=1} 2^A139352(k)/p^k) = 1.44736831993091923328... .

%t s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f[p_, e_] := 2^s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) s(e) = if(e > 3, s(e\4)) + e%4\2 \\ after _Charles R Greathouse IV_ at A139352

%o a(n) = vecprod(apply(x -> 2^s(x), factor(n)[, 2]));

%Y Cf. A037445, A139352, A366242, A366243, A366245, A366247, A366246, A366308.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Oct 06 2023