A066990 In canonical prime factorization of n replace even exponents with 2 and odd exponents with 1.

1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 6, 25, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 44, 45, 46, 47, 12, 49, 50, 51, 52, 53, 6, 55, 14, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69

Offset: 1

Author

Reinhard Zumkeller, Feb 01 2002

Keywords

• nonn
• nice
• mult
• easy

Comments

a(n) = n for cubefree numbers (A004709), whereas a(n) <> n for cube-full numbers (A046099).

The largest exponential divisor (A322791) of n that is cubefree (A004709). - Amiram Eldar, Jun 03 2025

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Brahim Mittou, New properties of an arithmetic function, Mathematica Montisnigri, Vol LIII (2022), pp. 5-11.
• Eric Weisstein's World of Mathematics, Cubefree.

Crossrefs

Cf. A004709, A046099, A322791.

Programs

Haskell

a066990 n = product $ zipWith (^)
           (a027748_row n) (map ((2 -) . (`mod` 2)) $ a124010_row n)
-- Reinhard Zumkeller, Dec 02 2012

Mathematica

fx[{a_,b_}]:={a,If[EvenQ[b],2,1]}; Table[Times@@(#[[1]]^#[[2]]&/@(fx/@ FactorInteger[n])),{n,70}] (* Harvey P. Dale, Jan 01 2012 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2 - f[i,2]%2));} \\ Amiram Eldar, Oct 28 2022

Python

from math import prod
from sympy import factorint
def a(n): return prod(p**(2-(e&1)) for p, e in factorint(n).items())
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, Jun 03 2025

Formula

Multiplicative with a(p^e) = p^(2 - e mod 2), p prime, e>0.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/30) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 0.4296463408... . - Amiram Eldar, Oct 28 2022