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A065883
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Remove factors of 4 from n (i.e., write n in base 4, drop final zeros, then rewrite in decimal).
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15
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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.
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
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EXAMPLE
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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.
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MAPLE
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A065883:= n -> n/4^floor(padic:-ordp(n, 2)/2):
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MATHEMATICA
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If[Divisible[#, 4], #/4^IntegerExponent[#, 4], #]&/@Range[80] (* Harvey P. Dale, Aug 31 2013 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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base,easy,nonn,mult
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AUTHOR
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STATUS
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approved
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