A375847 The maximum exponent in the prime factorization of the largest unitary cubefree divisor of n.

0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Aug 31 2024

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A004709, A036966, A038109, A051903, A330596, A337050, A360539.

Cf. A007424 (analogous with the largest cubefree divisor, for n >= 2).

Programs

Mathematica

a[n_] := Max[Join[{0}, Select[FactorInteger[n][[;; , 2]], # <= 2 &]]]; a[1] = 0; Array[a, 100]

PARI

a(n) = {my(e = select(x -> x <= 2, factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

Formula

a(n) = A051903(A360539(n)).

a(n) = 0 if and only if n is cubefull (A036966).

a(n) = 1 if and only if n is in A337050 \ A036966.

a(n) = 2 if and only if n is in A038109.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - A330596 = 1.25146474031763643535... .