A062565 Squarefree parts of 3-smooth numbers.

1, 2, 3, 1, 6, 2, 1, 3, 1, 2, 6, 3, 2, 1, 3, 6, 1, 2, 1, 6, 3, 2, 1, 2, 3, 6, 3, 1, 2, 1, 6, 3, 6, 2, 1, 2, 1, 3, 6, 3, 1, 2, 1, 2, 6, 3, 6, 2, 3, 1, 2, 1, 3, 6, 3, 1, 6, 2, 1, 2, 6, 1, 3, 6, 2, 3, 1, 2, 1, 3, 2, 6, 3, 1, 6, 2, 3, 1, 2, 6, 1, 3, 6, 2, 3, 1, 6, 2, 1, 3, 2, 6, 1, 3, 1, 6, 2, 3, 1

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

From Amiram Eldar, Jul 13 2023: (Start)

a(n) = A007913(A003586(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3. (End)

Example

3-smooth numbers are those expressible as 2^i*3^j, where i, j >= 0 (A003586). The tenth 3-smooth number is 18, in which the largest square is 9. 18/9 = 2, so a(10)=2.

Maple

N:= 10^20: # to use all 3-smooth numbers up to N
f:= proc(x) local a, b;
  a:= padic:-ordp(x, 2) mod 2; b:= padic:-ordp(x, 3) mod 2;
2^a*3^b
end proc:
map(f, sort([seq(seq(3^i*2^j, j=0..ilog2(N/3^i)), i=0..floor(log[3](N)))])); # Robert Israel, Dec 22 2017

Mathematica

Reap[Do[f = FactorInteger[n]; If[f[[-1, 1]] <= 3, Sow[Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]}& /@ f)]], {n, 10^5}]][[2, 1]] (* Jean-François Alcover, Feb 02 2018 *)

Crossrefs

Cf. A007913, A003586.

Sequence in context: A335195 A363266 A083855 * A175137 A156344 A218796

Adjacent sequences: A062562 A062563 A062564 * A062566 A062567 A062568

Keyword

nonn

Author

Neil Fernandez, Jul 03 2001

Extensions

More terms from Robert Israel, Dec 22 2017

Status

approved