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A212591
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a(n) is the smallest value of k for which A020986(k) = n.
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3
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0, 1, 2, 5, 8, 9, 10, 21, 32, 33, 34, 37, 40, 41, 42, 85, 128, 129, 130, 133, 136, 137, 138, 149, 160, 161, 162, 165, 168, 169, 170, 341, 512, 513, 514, 517, 520, 521, 522, 533, 544, 545, 546, 549, 552, 553, 554, 597, 640, 641, 642, 645, 648, 649, 650, 661
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OFFSET
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1,3
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COMMENTS
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Brillhart and Morton derive an omega function for the largest values of k. This sequence appears to be given by a similar alpha function.
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LINKS
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FORMULA
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a(2*n-1) - a(2*n-2) = (2^(2*k+1)+1)/3 and a(2*n) - a(2*n-1) = (2^(2*k+1)+1)/3 with a(0) = a(1) = 0, where n = (2^k)*(2*m-1) for some integers k >= 0 and m > 0.
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PROG
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(PARI)
alpha(n)={
if(n<2, return(max(0, n-1)));
local(nm1=n-1,
mi=m=ceil(nm1/2),
r=floor(log(m)/log(2)),
i, fi, alpha=0, a);
forstep(i=1, 2*r+1, 2,
mi/=2;
fi=(1+2^i)\3;
alpha+=fi*floor(0.5+mi);
);
alpha*=2;
if(nm1%2, \\ adjust for even n
a=factor(2*m)[1, 2]-1;
alpha-= (1+2^(1+2*a))\3;
);
return(alpha);
}
(J)
NB. J function on a vector
NB. Beware round-off errors on large arguments
NB. ok up to ~ 1e8
alphav =: 3 : 0
n =. <: y
if.+/ ntlo=. n > 0 do.
n =. ntlo#n
m =. >.-: n
r =. <.2^.m
f =. <.3%~2+2^2*>:i.>./>:r
z =. 0
mi =. m
for_i. i.#f do.
z =. z + (i{f) * <.0.5 + mi =. mi%2
end.
nzer=. (+/ @: (0=>./\)@:|.)"1 @: #: m
ntlo #^:_1 z - (2|n) * <.-:nzer{f
else.
ntlo
end.
)
NB. eg alphav 1 3 5 100 2 8 33
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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