A307848 The number of exponential infinitary divisors of n.

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, 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, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Amiram Eldar, May 01 2019

Keywords

• nonn
• mult

Comments

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.

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

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Andrew V. Lelechenko, Exponential and infinitary divisors, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; arXiv preprint, arXiv:1405.7597 [math.NT] (2014).

Crossrefs

Cf. A037445, A278908, A323308.

Programs

Mathematica

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 *)

Formula

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

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