A065883 Remove factors of 4 from n (i.e., write n in base 4, drop final zeros, then rewrite in decimal).
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, 7, 29, 30, 31, 2, 33, 34, 35, 9, 37, 38, 39, 10, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 54, 55, 14, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71, 18, 73, 74, 75
Offset: 1
Examples
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.
Links
- Indranil Ghosh, Table of n, a(n) for n = 1..20000 (First 1000 terms from Harry J. Smith)
Crossrefs
Programs
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Maple
A065883:= n -> n/4^floor(padic:-ordp(n,2)/2): map(A065883, [$1..1000]); # Robert Israel, Dec 08 2015
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Mathematica
If[Divisible[#,4],#/4^IntegerExponent[#,4],#]&/@Range[80] (* Harvey P. Dale, Aug 31 2013 *)
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PARI
a(n)=n/4^valuation(n,4); \\ Joerg Arndt, Dec 09 2015
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Python
def A065883(n): return n>>((~n&n-1).bit_length()&-2) # Chai Wah Wu, Jul 09 2022
Formula
If n mod 4 = 0 then a(n) = a(n/4), otherwise a(n) = n.
Multiplicative with a(p^e) = 2^(e (mod 2)) if p = 2 and a(p^e) = p^e if p is an odd prime.
a(n) = n/4^A235127(n).
a(n) = A214392(n) if n mod 16 != 0. - Peter Kagey, Sep 02 2015
From Robert Israel, Dec 08 2015: (Start)
G.f.: x/(1-x)^2 - 3 Sum_{j>=1} x^(4^j)/(1-x^(4^j))^2.
G.f. satisfies G(x) = G(x^4) + x/(1-x)^2 - 4 x^4/(1-x^4)^2. (End)
Sum_{k=1..n} a(k) ~ (2/5) * n^2. - Amiram Eldar, Nov 20 2022
Dirichlet g.f.: zeta(s-1)*(4^s-4)/(4^s-1). - Amiram Eldar, Jan 04 2023