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%I #65 Jan 04 2023 02:10:01
%S 1,2,3,1,5,6,7,2,9,10,11,3,13,14,15,1,17,18,19,5,21,22,23,6,25,26,27,
%T 7,29,30,31,2,33,34,35,9,37,38,39,10,41,42,43,11,45,46,47,3,49,50,51,
%U 13,53,54,55,14,57,58,59,15,61,62,63,1,65,66,67,17,69,70,71,18,73,74,75
%N Remove factors of 4 from n (i.e., write n in base 4, drop final zeros, then rewrite in decimal).
%H Indranil Ghosh, <a href="/A065883/b065883.txt">Table of n, a(n) for n = 1..20000</a> (First 1000 terms from Harry J. Smith)
%F If n mod 4 = 0 then a(n) = a(n/4), otherwise a(n) = n.
%F Multiplicative with a(p^e) = 2^(e (mod 2)) if p = 2 and a(p^e) = p^e if p is an odd prime.
%F a(n) = n/4^A235127(n).
%F a(n) = A214392(n) if n mod 16 != 0. - _Peter Kagey_, Sep 02 2015
%F From _Robert Israel_, Dec 08 2015: (Start)
%F G.f.: x/(1-x)^2 - 3 Sum_{j>=1} x^(4^j)/(1-x^(4^j))^2.
%F G.f. satisfies G(x) = G(x^4) + x/(1-x)^2 - 4 x^4/(1-x^4)^2. (End)
%F Sum_{k=1..n} a(k) ~ (2/5) * n^2. - _Amiram Eldar_, Nov 20 2022
%F Dirichlet g.f.: zeta(s-1)*(4^s-4)/(4^s-1). - _Amiram Eldar_, Jan 04 2023
%e a(7)=7, a(14)=14, a(28)=a(4*7)=7, a(56)=a(4*14)=14, a(112)=a(4^2*7)=7.
%p A065883:= n -> n/4^floor(padic:-ordp(n,2)/2):
%p map(A065883, [$1..1000]); # _Robert Israel_, Dec 08 2015
%t If[Divisible[#,4],#/4^IntegerExponent[#,4],#]&/@Range[80] (* _Harvey P. Dale_, Aug 31 2013 *)
%o (PARI) baseA2B(x, a, b)= { local(d, e=0, f=1); while (x>0, d=x%b; x\=b; e+=d*f; f*=a); return(e) }
%o { for (n=1, 1000, if (n%4, a=n, a=baseA2B(n, 10, 4); while (a%10 == 0, a\=10); a=baseA2B(a, 4, 10)); write("b065883.txt", n, " ", a) ) } \\ _Harry J. Smith_, Nov 03 2009
%o (PARI) a(n)=n/4^valuation(n,4); \\ _Joerg Arndt_, Dec 09 2015
%o (Python)
%o def A065883(n): return n>>((~n&n-1).bit_length()&-2) # _Chai Wah Wu_, Jul 09 2022
%Y Cf. A214392, A235127, A350091 (drop final 2's).
%Y Remove other factors: A000265, A038502, A132739, A244414, A242603, A004151.
%K base,easy,nonn,mult
%O 1,2
%A _Henry Bottomley_, Nov 26 2001
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