A375359 The maximum exponent in the prime factorization of the smallest number whose square is divisible by n.

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

Offset

1,8

Comments

Differs from A050361 at n = 1, 64, 128, 192, ... .

Differs from A366902 at n = 1, 64, 192, 216, ... .

Differs from A325837 at n = 1, 216, 432, 648, ... .

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(A019554(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum{k>=1} (1 - 1/zeta(2*k+1)) = 1.21464720975357037829... .

From Amiram Eldar, Nov 17 2025: (Start)

a(n) = A110654(A051903(n)) = ceiling(A051903(n)/2).

a(n) = A051903(A053143(n))/2 (half the maximum exponent in the prime factorization of the smallest square divisible by n). (End)

Mathematica

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[(If[EvenQ[#], #, # + 1]) & /@ e]/2]; a[1] = 0; Array[a, 100]

Prog

(PARI) a(n) = if(n == 1, 0, vecmax(apply(x -> if(x % 2, x+1, x), factor(n)[, 2]))/2);

Crossrefs

Cf. A019554, A051903, A053143, A110654, A372602, A375360.

Cf. A050361, A325837, A366902.

Sequence in context: A385418 A061704 A325837 * A366902 A381715 A050361

Adjacent sequences: A375356 A375357 A375358 * A375360 A375361 A375362

Keyword

nonn,easy

Author

Amiram Eldar, Aug 13 2024

Status

approved