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

%I #28 Oct 28 2022 07:14:58

%S 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,

%T 28,29,30,31,2,33,34,35,36,37,38,39,10,41,42,43,44,45,46,47,12,49,50,

%U 51,52,53,6,55,14,57,58,59,60,61,62,63,4,65,66

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

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

%H Reinhard Zumkeller, <a href="/A066990/b066990.txt">Table of n, a(n) for n = 1..10000</a>

%H Brahim Mittou, <a href="https://www.montis.pmf.ac.me/allissues/53/Mathematica-Montisnigri-53-1.pdf">New properties of an arithmetic function</a>, Mathematica Montisnigri, Vol LIII (2022).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cubefree.html">Cubefree</a>.

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

%F 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

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

%o (Haskell)

%o a066990 n = product $ zipWith (^)

%o (a027748_row n) (map ((2 -) . (`mod` 2)) $ a124010_row n)

%o -- _Reinhard Zumkeller_, Dec 02 2012

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

%Y Cf. A004709, A046099.

%K nonn,nice,mult

%O 1,2

%A _Reinhard Zumkeller_, Feb 01 2002