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

1,9

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.

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A056975 A370482 A294079 * A114117 A144435 A182533

Adjacent sequences: A375666 A375667 A375668 * A375670 A375671 A375672

Keyword

nonn,easy

Author

Amiram Eldar, Aug 23 2024

Status

approved