%I #15 Nov 08 2020 05:57:09

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

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

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

%N The number of exponential infinitary divisors of n.

%C The exponential infinitary divisors of Product p(i)^r(i) are all the numbers of the form Product p(i)^s(i) where s(i) if an infinitary divisor of r(i) for all i.

%C Differs from A278908 at n = 256, 768, 1280, 1792, 2304, 2816, ...

%C Differs from A323308 at n = 64, 192, 256, 320, 448, 576, 704, ...

%H Amiram Eldar, <a href="/A307848/b307848.txt">Table of n, a(n) for n = 1..10000</a>

%H Andrew V. Lelechenko, <a href="https://doi.org/10.1007/s11253-017-1289-7">Exponential and infinitary divisors</a>, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; <a href="https://arxiv.org/abs/1405.7597">arXiv preprint</a>, arXiv:1405.7597 [math.NT] (2014).

%F Multiplicative with a(p^e) = A037445(e).

%F Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.5482125828..., where d(k) = A037445(k). - _Amiram Eldar_, Nov 08 2020

%t di[1] = 1; di[n_] := Times @@ Flatten[ 2^DigitCount[#, 2, 1]& /@ FactorInteger[n][[All, 2]] ]; fun[p_,e_] := di[e]; a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100] (* after _Jean-François Alcover_ at A037445 *)

%Y Cf. A037445, A278908, A323308.

%K nonn,mult

%O 1,4

%A _Amiram Eldar_, May 01 2019