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

1,4

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).

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

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 *)

Crossrefs

Cf. A037445, A278908, A323308.

Sequence in context: A323308 A365549 A278908 * A358260 A368978 A381613

Adjacent sequences: A307845 A307846 A307847 * A307849 A307850 A307851

Keyword

nonn,mult

Author

Amiram Eldar, May 01 2019

Status

approved