A383009 -id:A383009 - cp's OEIS frontend

A383004 Exponent of the highest power of 2 dividing the n-th cubefree number.

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

Offset: 1

Amiram Eldar, Apr 12 2025

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

Cf. A004709, A007814, A373550, A383005, A383009.

cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; IntegerExponent[Select[Range[200], cubeFreeQ], 2]

iscubefree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 2, return (0))); 1; }
list(lim) = for(k = 1, lim, if(iscubefree(k), print1(valuation(k, 2), ", ")));

a(n) = A007814(A004709(n)).
A383009(a(n)) > 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4/7.

A383008 Indices of the even terms in the sequence of squarefree numbers.

Original entry on oeis.org

2, 5, 7, 10, 15, 17, 19, 22, 25, 28, 30, 36, 39, 41, 44, 47, 49, 51, 54, 59, 63, 66, 69, 72, 74, 76, 80, 83, 85, 88, 91, 94, 97, 102, 104, 106, 108, 111, 114, 116, 119, 124, 127, 129, 132, 135, 138, 140, 143, 148, 151, 156, 159, 161, 164, 169, 171, 173, 176, 178

Offset: 1

Amiram Eldar, Apr 12 2025

The asymptotic density of this sequence is 1/3.

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

Cf. A005117, A039956, A071403, A373550, A383009.

Position[Select[Range[350], SquareFreeQ], _?EvenQ] // Flatten

list(lim) = {my(c = 0); for(k = 1, lim, if(issquarefree(k), c++; if(!(k % 2), print1(c, ", "))));}

A005117(a(n)) = A039956(n).

A373550(a(n)) = 0.

cp's OEIS Frontend

A383004 Exponent of the highest power of 2 dividing the n-th cubefree number.

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