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A124508 a(n) = 2^BigO(n) * 3^omega(n), where BigO = A001222 and omega = A001221, the numbers of prime factors of n with and without repetitions.

%I A124508 #16 Feb 16 2025 08:33:03
%S A124508 1,6,6,12,6,36,6,24,12,36,6,72,6,36,36,48,6,72,6,72,36,36,6,144,12,36,
%T A124508 24,72,6,216,6,96,36,36,36,144,6,36,36,144,6,216,6,72,72,36,6,288,12,
%U A124508 72,36,72,6,144,36,144,36,36,6,432,6,36,72,192,36,216,6,72,36,216,6,288,6
%N A124508 a(n) = 2^BigO(n) * 3^omega(n), where BigO = A001222 and omega = A001221, the numbers of prime factors of n with and without repetitions.
%H A124508 Reinhard Zumkeller, <a href="/A124508/b124508.txt">Table of n, a(n) for n = 1..10000</a>
%H A124508 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactorization.html">Prime Factorization</a>.
%H A124508 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth number</a>.
%F A124508 Multiplicative with p^e -> 3*2^e, p prime and e>0.
%F A124508 a(n) = A061142(n)*A074816(n) = A000079(A001222(n))*A000244(A001221(n)).
%F A124508 A124509 gives the range: A124509(n) = a(A124510(n)) and a(m) <> a(A124510(n)) for m < A124510(n).
%F A124508 For primes p, q with p <> q: a(p) = 6; a(p*q) = 36; a(p^k) = 3*2^k, k>0.
%F A124508 For squarefree numbers m: a(m) = 6^omega(m).
%F A124508 A001222(a(n)) = A001222(n)+1; A001221(a(n)) = 2 for n > 1.
%F A124508 A124511(n) = a(a(n)); A124512(n) = a(a(a(n))).
%t A124508 Table[2^PrimeOmega[n] 3^PrimeNu[n],{n,80}] (* _Harvey P. Dale_, Mar 26 2013 *)
%o A124508 (PARI) a(n) = my(f = factor(n)); 2^bigomega(f) * 3^omega(f); \\ _Amiram Eldar_, Jul 11 2023
%Y A124508 Cf. A000079, A000244, A000400, A001221, A001222, A003586, A005117, A007283, A061142, A074816, A124509, A124510, A124511, A124512.
%K A124508 nonn,easy,mult
%O A124508 1,2
%A A124508 _Reinhard Zumkeller_, Nov 04 2006