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A065883 Remove factors of 4 from n (i.e., write n in base 4, drop final zeros, then rewrite in decimal). 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Indranil Ghosh, Table of n, a(n) for n = 1..20000 (First 1000 terms from Harry J. Smith)
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
EXAMPLE
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.
MAPLE
A065883:= n -> n/4^floor(padic:-ordp(n, 2)/2):
map(A065883, [$1..1000]); # Robert Israel, Dec 08 2015
MATHEMATICA
If[Divisible[#, 4], #/4^IntegerExponent[#, 4], #]&/@Range[80] (* Harvey P. Dale, Aug 31 2013 *)
PROG
(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) }
{ 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
(PARI) a(n)=n/4^valuation(n, 4); \\ Joerg Arndt, Dec 09 2015
(Python)
def A065883(n): return n>>((~n&n-1).bit_length()&-2) # Chai Wah Wu, Jul 09 2022
CROSSREFS
Cf. A214392, A235127, A350091 (drop final 2's).
Remove other factors: A000265, A038502, A132739, A244414, A242603, A004151.
Sequence in context: A083346 A319652 A327938 * A214392 A071975 A350389
KEYWORD
base,easy,nonn,mult
AUTHOR
Henry Bottomley, Nov 26 2001
STATUS
approved

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Last modified April 20 06:37 EDT 2024. Contains 371799 sequences. (Running on oeis4.)