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%I #30 Oct 30 2022 05:27:25
%S 1,2,3,1,5,6,7,4,1,10,11,3,13,14,15,1,17,2,19,5,21,22,23,12,1,26,9,7,
%T 29,30,31,8,33,34,35,1,37,38,39,20,41,42,43,11,5,46,47,3,1,2,51,13,53,
%U 18,55,28,57,58,59,15,61,62,7,1,65,66,67,17,69,70,71,4,73,74,3,19,77
%N Denominator of rational number i/j such that Sagher map sends i/j to n.
%C The Sagher map sends Product p_i^e_i / Product q_i^f_i (p_i and q_i being distinct primes) to Product p_i^(2e_i) * Product q_i^(2f_i-1). This is multiplicative.
%H Reinhard Zumkeller, <a href="/A071975/b071975.txt">Table of n, a(n) for n = 1..10000</a>
%H Yoram Sagher, <a href="http://www.jstor.org/stable/2324846">Counting the rationals</a>, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.
%F If n = Product p_i^e_i, then a(n) = Product p_i^f(e_i), where f(n) = (n+1)/2 if n is odd and f(n) = 0 if n is even. - _Reiner Martin_, Jul 08 2002
%F From _Reinhard Zumkeller_, Jul 10 2011: (Start)
%F a(n^2) = 1, A071974(n^2) = n, cf. A000290.
%F a(2*(2*n-1)^2) = 2, A071974(2*(2*n-1)^2) = 2*n+1, cf. A077591.
%F a(2*(2*n-1)^2) = 2, A071974(2*(2*n-1)^2) = 2*n+1, cf. A077591. (End)
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4*zeta(3)/180) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 - 1/p^5 + 1/p^6) = 0.3394877587... . - _Amiram Eldar_, Oct 30 2022
%e The Sagher map sends the following fractions to 1, 2, 3, 4, ...: 1/1, 1/2, 1/3, 2/1, 1/5, 1/6, 1/7, 1/4, 3/1, ...
%t f[{p_, a_}] := If[OddQ[a], p^((a+1)/2), 1]; a[n_] := Times@@(f/@FactorInteger[n])
%o (PARI) a(n)=local(v=factor(n)~); prod(k=1,length(v),if(v[2,k]%2,v[1,k]^-(-v[2,k]\2),1))
%o (Haskell)
%o a071975 n = product $ zipWith (^) (a027748_row n) $
%o map (\e -> (e `mod` 2) * (e + 1) `div` 2) $ a124010_row n
%o -- _Reinhard Zumkeller_, Jun 15 2012
%Y Cf. A000290, A071974.
%Y Cf. A002117, A027748, A124010.
%K nonn,frac,easy,nice,mult
%O 1,2
%A _N. J. A. Sloane_, Jun 19 2002
%E More terms from _Reiner Martin_, Jul 08 2002
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