A375670 The maximum exponent in the prime factorization of the largest 5-rough divisor of n.

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

Offset

1,25

Comments

First differs from A106799 at n = 35.

The largest exponent among the exponents of the primes that are larger than 3 in the prime factorization of n.

Links

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

Index entries for sequences computed from exponents in factorization of n.

Formula

a(n) = A051903(A065330(n)).

a(n) = 0 if and only if n is a 3-smooth number (A003586).

a(n) = 1 if and only if n is a product of a squarefree 5-rough number larger than 1 and a 3-smooth number.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) = 1.1034178389191320571029... , where d(k) is the asymptotic density of the occurrences of k in this sequence: d(1) = 3/(2*zeta(2)), and d(k) = (1/zeta(k+1)) / ((1-1/2^(k+1))*(1-1/3^(k+1))) - (1/zeta(k)) / ((1-1/2^k)*(1-1/3^k)) for k >= 2.

In general, the asymptotic mean of the maximum exponent in the prime factorization of the largest p-rough divisor of n is Sum_{k>=1} k * d(k), where d(1) = 1/(zeta(2) * f(p, 2)), d(k) = 1/(zeta(k+1) * f(p, k+1)) - 1/(zeta(k) * f(p, k)) for k >= 2, and f(p, m) = Product_{q prime < p} (1-1/q^m).

Mathematica

a[n_] := Module[{m = n / Times@@({2, 3}^IntegerExponent[n, {2, 3}])}, If[m == 1, 0, Max[FactorInteger[m][[;; , 2]]]]]; Array[a, 100]

Prog

(PARI) a(n) = {my(m = n >> valuation(n, 2)/3^valuation(n, 3)); if(m == 1, 0, vecmax(factor(m)[, 2])); }

Crossrefs

Cf. A003586, A007310 (5-rough numbers), A051903, A065330, A106799, A375669.

Sequence in context: A037853 A255237 A291954 * A339885 A106799 A212210

Adjacent sequences: A375667 A375668 A375669 * A375671 A375672 A375673

Keyword

nonn,easy

Author

Amiram Eldar, Aug 23 2024

Status

approved