A375033 The maximum even exponent in the prime factorization of n, or 0 if no such exponent exists.

0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0

Offset

1,4

Comments

First differs from A350386 at n = 36.

The asymptotic density of the occurrences of 0's is d(0) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463; the asymptotic density of the exponentially odd numbers, A268335).

The asymptotic density of the occurrences of 2*k, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+1)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k-1)*(p+1))).

For example, the asymptotic density of the occurrences of 2's is d(1) = Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = 0.243291... (the asymptotic density of A375031).

Links

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

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

Formula

max(a(n), A375032(n)) = A051903(n).

a(n) = 0 if and only if n is an exponentially odd number (A268335).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (2*k) * d(k) = 0.72584606502990528747..., where d(k) is defined in the Comments section above.

a(n) = A051903(A350388(n)). - Amiram Eldar, Aug 17 2024

Mathematica

a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], EvenQ]]]; a[1] = 0; Array[a, 100]

Prog

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

Crossrefs

Cf. A051903, A065463, A162642, A268335, A350386, A350388, A375031, A375032.

Sequence in context: A171919 A000089 A339430 * A350386 A352550 A348326

Adjacent sequences: A375030 A375031 A375032 * A375034 A375035 A375036

Keyword

nonn,easy

Author

Amiram Eldar, Jul 28 2024

Status

approved