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

Author

Amiram Eldar, Nov 08 2020

Keywords

• nonn
• mult

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.

Crossrefs

Cf. A005117, A007947, A209061, A278908.

Programs

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]

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).