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

%I A066990 #36 Jun 03 2025 11:25:56
%S A066990 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 A066990 28,29,30,31,2,33,34,35,36,37,38,39,10,41,42,43,44,45,46,47,12,49,50,
%U A066990 51,52,53,6,55,14,57,58,59,60,61,62,63,4,65,66,67,68,69
%N A066990 In canonical prime factorization of n replace even exponents with 2 and odd exponents with 1.
%C A066990 a(n) = n for cubefree numbers (A004709), whereas a(n) <> n for cube-full numbers (A046099).
%C A066990 The largest exponential divisor (A322791) of n that is cubefree (A004709). - _Amiram Eldar_, Jun 03 2025
%H A066990 Reinhard Zumkeller, <a href="/A066990/b066990.txt">Table of n, a(n) for n = 1..10000</a>
%H A066990 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), pp. 5-11.
%H A066990 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cubefree.html">Cubefree</a>.
%F A066990 Multiplicative with a(p^e) = p^(2 - e mod 2), p prime, e>0.
%F A066990 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 A066990 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 A066990 (Haskell)
%o A066990 a066990 n = product $ zipWith (^)
%o A066990            (a027748_row n) (map ((2 -) . (`mod` 2)) $ a124010_row n)
%o A066990 -- _Reinhard Zumkeller_, Dec 02 2012
%o A066990 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2 - f[i,2]%2));} \\ _Amiram Eldar_, Oct 28 2022
%o A066990 (Python)
%o A066990 from math import prod
%o A066990 from sympy import factorint
%o A066990 def a(n): return prod(p**(2-(e&1)) for p, e in factorint(n).items())
%o A066990 print([a(n) for n in range(1, 70)]) # _Michael S. Branicky_, Jun 03 2025
%Y A066990 Cf. A004709, A046099, A322791.
%K A066990 nonn,nice,mult,easy
%O A066990 1,2
%A A066990 _Reinhard Zumkeller_, Feb 01 2002