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A307848
The number of exponential infinitary divisors of n.
7
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
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
KEYWORD
nonn,mult
AUTHOR
Amiram Eldar, May 01 2019
STATUS
approved