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A065338 a(1) = 1, a(p) = p mod 4 for p prime and a(u * v) = a(u) * a(v) for u, v > 0.

%I A065338 #29 Sep 02 2023 04:38:06
%S A065338 1,2,3,4,1,6,3,8,9,2,3,12,1,6,3,16,1,18,3,4,9,6,3,24,1,2,27,12,1,6,3,
%T A065338 32,9,2,3,36,1,6,3,8,1,18,3,12,9,6,3,48,9,2,3,4,1,54,3,24,9,2,3,12,1,
%U A065338 6,27,64,1,18,3,4,9,6,3,72,1,2,3,12,9,6,3,16,81,2,3,36,1,6,3,24,1,18,3
%N A065338 a(1) = 1, a(p) = p mod 4 for p prime and a(u * v) = a(u) * a(v) for u, v > 0.
%H A065338 T. D. Noe, <a href="/A065338/b065338.txt">Table of n, a(n) for n = 1..1000</a>
%F A065338 a(n) = 1 if n = 1, otherwise (A020639(n) mod 4) * n / A020639(n).
%F A065338 a(n) = (2^A007814(n)) * (3^A065339(n)).
%F A065338 a(n) <= n.
%F A065338 a(a(n)) = a(n).
%F A065338 a(x) = x iff x = 2^i * 3^j for i, j >= 0.
%F A065338 a(A003586(n)) = A003586(n).
%F A065338 a(A065331(n)) = A065331(n).
%F A065338 a(A004613(n)) = 1; A065333(a(n)) = 1. - _Reinhard Zumkeller_, Jul 10 2010
%F A065338 Dirichlet g.f.: (1/(1-2^(-s+1))) * Product_{prime p=4k+1} (1/(1-p^(-s))) * Product_{prime p=4k+3} 1/(1-3*p^(-s)). - _Ralf Stephan_, Mar 28 2015
%e A065338 a(120) = a(2*2*2*3*5) = a(2)*a(2)*a(2)*a(3)*a(5) = 2*2*2*3*1 = 24.
%e A065338 a(150) = a(2*3*5*5) = a(2)*a(3)*a(5)*a(5) = 2*3*1*1 = 6.
%e A065338 a(210) = a(2*3*5*7) = a(2)*a(3)*a(5)*a(7) = 2*3*1*3 = 18.
%t A065338 a[1] = 1; a[n_] := a[n] = Mod[p = FactorInteger[n][[1, 1]], 4]*a[n/p]; Table[ a[n], {n, 1, 100} ] (* _Jean-François Alcover_, Jan 20 2012 *)
%o A065338 (Haskell)
%o A065338 a065338 1 = 1
%o A065338 a065338 n = (spf `mod` 4) * a065338 (n `div` spf) where spf = a020639 n
%o A065338 -- _Reinhard Zumkeller_, Nov 18 2011
%o A065338 (PARI) a(n)=my(f=factor(n)); prod(i=1,#f~, (f[i,1]%4)^f[i,2]) \\ _Charles R Greathouse IV_, Feb 07 2017
%Y A065338 Cf. A039702, A000040, A003586, A007814, A065339, A065331.
%K A065338 mult,nice,nonn
%O A065338 1,2
%A A065338 _Reinhard Zumkeller_, Oct 29 2001