A383006 Exponent of the highest power of 2 dividing the n-th powerful number.

0, 2, 3, 0, 4, 0, 0, 5, 2, 0, 6, 3, 0, 2, 2, 0, 0, 7, 4, 0, 2, 3, 3, 0, 0, 8, 5, 0, 2, 0, 0, 3, 4, 4, 0, 2, 2, 9, 0, 6, 0, 3, 0, 2, 0, 4, 5, 0, 5, 2, 0, 3, 2, 3, 10, 0, 0, 7, 2, 0, 4, 0, 0, 3, 0, 2, 2, 0, 5, 6, 0, 6, 2, 3, 0, 4, 3, 4, 0, 11, 2, 0, 0, 0, 8, 3, 0

Offset: 1

Author

Amiram Eldar, Apr 12 2025

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to powerful numbers.

Crossrefs

Cf. A001694, A007814, A363189, A373549, A383007.

Programs

Mathematica

IntegerExponent[Select[Range[5000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], 2]

PARI

list(lim) = for(k = 1, lim, if(ispowerful(k), print1(valuation(k, 2), ", ")));

Formula

a(n) = A007814(A001694(n)).

a(A363189(n)) = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (11 + 6*sqrt(2))/7 = 2.7836116... .