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

Offset

1,2

Comments

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

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).

Eric Weisstein's World of Mathematics, Cubefree.

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

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 *)

Prog

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

Crossrefs

Cf. A004709, A046099.

Sequence in context: A043267 A167514 A091733 * A365296 A097449 A104415

Adjacent sequences: A066987 A066988 A066989 * A066991 A066992 A066993

Keyword

nonn,nice,mult

Author

Reinhard Zumkeller, Feb 01 2002

Status

approved