A338782 The largest e-squarefree e-divisor of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 12, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset

1,2

Comments

An exponential squarefree exponential divisor (e-squarefree e-divisor) d = Product p_i^f_i of n = Product p_i^e_i has f_i | e_i and f_i is squarefree for all i.

The largest of the A278908(n) e-squarefree e-divisors of n.

a(n) = n if and only if n is an exponentially squarefree number (A209061).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166, arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Formula

Multiplicative with a(p^e) = p^rad(e), where rad(k) is the largest squarefree number dividing k (A007947).

Sum_{n<=x} a(n) = (1/2) * c * x^2, where c = Product_{p prime} Sum{k>=4} (p^rad(k) - p^(1+rad(k-1)))/p^(2*k) = 0.9646498658... (Tóth, 2007).

Mathematica

rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); f[p_, e_] := p^rad[e]; a[1]=1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Crossrefs

Cf. A005117, A007947, A209061, A278908.

Sequence in context: A056962 A043275 A216455 * A291580 A291581 A058035

Adjacent sequences: A338779 A338780 A338781 * A338783 A338784 A338785

Keyword

nonn,mult

Author

Amiram Eldar, Nov 08 2020

Status

approved