A375669 The maximum exponent in the prime factorization of the largest odd divisor of n.

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Aug 23 2024

Keywords

• nonn
• easy

Comments

The largest exponent among the exponents of the odd primes 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.

Crossrefs

Cf. A000079, A000265, A051903, A122132, A375670.

Programs

Mathematica

a[n_] := Module[{o = n / 2^IntegerExponent[n, 2]}, If[o == 1, 0, Max[FactorInteger[o][[;;, 2]]]]]; Array[a, 100]

PARI

a(n) = {my(o = n >> valuation(n, 2)); if(o == 1, 0, vecmax(factor(o)[,2]));}

Formula

a(n) = A051903(A000265(n)).

a(n) = 0 if and only if n is a power of 2 (A000079).

a(n) = 1 if and only if n is in A122132 \ A000079.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) = 1.25979668632898014495... , where d(k) is the asymptotic density of the occurrences of k in this sequence: d(1) = 4/(3*zeta(2)), and d(k) = (1/zeta(k+1)) / (1-1/2^(k+1)) - (1/zeta(k)) / (1-1/2^k) for k >= 2.