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A160467
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a(n) = 1 if n is odd; otherwise, a(n) = 2^(k-1) where 2^k is the largest power of 2 that divides n.
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5
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1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 16, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 32, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 16
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OFFSET
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1,4
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COMMENTS
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Fifth factor of the row sums A160466 of the Eta triangle A160464.
From Peter Luschny, May 31 2009: (Start)
Let odd(n) be the characteristic function of the odd numbers (A000035) and sigma(n) the number of 1's in binary expansion of n (A000120). Then a(n) = 2^(sigma(n-1) - sigma(n) + odd(n)).
Let B_{n} be the Bernoulli number. Then this sequence is also
a(n) = denominator(4*(4^n-1)*B_{2*n}/n). (End)
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..16384
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FORMULA
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a(n) = A026741(n)/A000265(n). - Paul Curtz, Apr 18 2010
a(n) = 2^max(A007814(n) - 1, 0). - Max Alekseyev, Feb 09 2011
a((2*n-1)*2^p) = A011782(p), p >= 0 and n >= 1. - Johannes W. Meijer, Jan 25 2013
a(n) = (1 + A140670(n))/2. - Antti Karttunen, Nov 18 2017
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MAPLE
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nmax:=96: p:= floor(log[2](nmax)): for n from 1 to nmax do a(n):=1 end do: for q from 1 to p do for n from 1 to nmax do if n mod 2^q = 0 then a(n):= 2^(q-1) end if: end do: end do: seq(a(n), n=1..nmax);
From Peter Luschny, May 31 2009: (Start)
a := proc(n) local sigma; sigma := proc(n) local i; add(i, i=convert(n, base, 2)) end; 2^(sigma(n-1)-sigma(n)+`if`(type(n, odd), 1, 0)) end: seq(a(n), n=1..96);
a := proc(n) denom(4*(4^n-1)*bernoulli(2*n)/n) end: seq(a(n), n=1..96); (End)
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MATHEMATICA
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a[n_] := If[OddQ[n], 1, 2^(IntegerExponent[n, 2] - 1)]; Array[a, 100] (* Amiram Eldar, Jul 02 2020 *)
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PROG
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(PARI) A160467(n) = 2^max(valuation(n, 2)-1, 0); \\ Antti Karttunen, Nov 18 2017, after Max Alekseyev's Feb 09 2011 formula.
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CROSSREFS
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Cf. A000265, A007814, A026741, A140670, A160464, A220466.
Sequence in context: A085384 A067856 A343370 * A345000 A122374 A261960
Adjacent sequences: A160464 A160465 A160466 * A160468 A160469 A160470
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KEYWORD
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base,easy,nonn,mult
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AUTHOR
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Johannes W. Meijer, May 24 2009, Jun 28 2011
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EXTENSIONS
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Keyword mult added by Max Alekseyev, Feb 09 2011
Name changed by Antti Karttunen, Nov 18 2017
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STATUS
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approved
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