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A046643 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.

%I A046643 #34 May 04 2025 10:15:50
%S A046643 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,
%T A046643 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,
%U A046643 1,3,1,1,1,15,1,1,3,3,1,1,1,35,35,1,1,3,1,1,1,5,1,3
%N A046643 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.
%C A046643 b(n) = A046643(n)/A046644(n) is multiplicative with b(p^n) = (2n-1)!!/2^n/n!. Dirichlet g.f. of A046643(n)/A046644(n) is sqrt(zeta(x)). - _Christian G. Bower_, May 16 2005
%C A046643 That is, b(p^n) = A001147(n) / (A000079(n)*A000142(n)) = A010050(n)/A000290(A000165(n)) = (2n)!/((2^n*n!)^2). - _Antti Karttunen_, Jul 08 2017
%H A046643 G. C. Greubel, <a href="/A046643/b046643.txt">Table of n, a(n) for n = 1..10000</a>
%H A046643 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A046643 Sum_{b|d} b(d)b(n/d) = 1. Also b_{2^j} = A001790[ j ]/2^A005187[ j ].
%F A046643 From _Antti Karttunen_, Jul 08 2017: (Start)
%F A046643 Multiplicative with a(p^n) = A001790(n).
%F A046643 a(1) = 1; for n > 1, a(n) = A001790(A067029(n)) * a(A028234(n)).
%F A046643 (End)
%F A046643 Sum_{j=1..n} A046643(j)/A046644(j) ~ n / sqrt(Pi*log(n)) * (1 + (1 - gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, May 04 2025
%e A046643 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, ...
%p A046643 b := proc(n) option remember; local c,i,t1; if n = 1 then 1 else c := 1; t1 := divisors(n);
%p A046643 for i from 2 to nops(t1)-1 do c := c-b(t1[ i ])*b(n/t1[ i ]); od; c/2; fi; end;
%t A046643 b[1] = 1; b[n_] := b[n] = (dn = Divisors[n]; c = 1;
%t A046643 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 *)
%o A046643 (PARI)
%o A046643 A046643perA046644(n) = { my(c=1); if(1==n,c,fordiv(n,d, if((d>1)&&(d<n), c -= (A046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); }
%o A046643 A046643(n) = numerator(A046643perA046644(n)); \\ After Maple-program, _Antti Karttunen_, Jul 08 2017
%o A046643 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 04 2025
%o A046643 (Scheme)
%o A046643 (define (A046643 n) (if (= 1 n) n (* (A001790 (A067029 n)) (A046643 (A028234 n)))))
%o A046643 ;; Or, after _Christian G. Bower_'s May 16 2005 comment:
%o A046643 (definec (A046643perA046644 n) (if (= 1 n) n (* (/ (A010050 (A067029 n)) (A000290 (A000165 (A067029 n)))) (A046643perA046644 (A028234 n)))))
%o A046643 (define (A046643 n) (numerator (A046643perA046644 n)))
%o A046643 (define (A046644 n) (denominator (A046643perA046644 n)))
%o A046643 ;; _Antti Karttunen_, Jul 08 2017
%Y A046643 Cf. A000079, A000165, A001147, A001790, A005187, A010050, A028234, A067029.
%Y A046643 Cf. A046644, A046645.
%K A046643 nonn,easy,frac,nice,mult
%O A046643 1,4
%A A046643 _N. J. A. Sloane_, Dec 11 1999