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A046643
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From square root of Riemann zeta function: form Dirichlet series Sum b_n/n^s whose square is zeta function; sequence gives numerator of b_n.
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4
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1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 35, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 63, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 35, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 231, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 35, 35, 1, 1, 3, 1, 1, 1, 5, 1, 3
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OFFSET
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1,4
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COMMENTS
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A046643(n)/A046644(n) is multiplicative with a(p^e) = (2n-1)!!/2^n/n!. Dirichlet g.f. of A046643(n)/A046644(n) is sqrt(zeta(x)). Christian G. Bower May 16, 2005.
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LINKS
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Table of n, a(n) for n=1..90.
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FORMULA
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Sum_{b|d} b(d)b(n/d) = 1. Also b_{2^j} = A001790[ j ]/2^A005187[ j ].
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EXAMPLE
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b_1, b_2, ... = 1, 1/2, 1/2, 3/8, 1/2, 1/4, 1/2, 5/16, 3/8, 1/4, 1/2, 3/16, ...
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MAPLE
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b := proc(n) option remember; local c, i, t1; if n = 1 then 1 else c := 1; t1 := divisors(n);
for i from 2 to nops(t1)-1 do c := c-b(t1[ i ])*b(n/t1[ i ]); od; c/2; fi; end;
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MATHEMATICA
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b[1] = 1; b[n_] := b[n] = (dn = Divisors[n]; c = 1;
Do[c = c - b[dn[[i]]]*b[n/dn[[i]]], {i, 2, Length[dn] - 1}]; c/2); a[n_] := Numerator[b[n]]; a /@ Range[90] (* Jean-François Alcover, Apr 04 2011 , after Maple version *)
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CROSSREFS
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Sequence in context: A081060 A131268 A109221 * A112475 A126799 A135494
Adjacent sequences: A046640 A046641 A046642 * A046644 A046645 A046646
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KEYWORD
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nonn,easy,frac,nice,mult
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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