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A071974 Numerator of rational number i/j such that Sagher map sends i/j to n. 7
1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
LINKS
David M. Bradley, Counting the Positive Rationals: A Brief Survey, arXiv:math/0509025 [math.HO], 2005.
Gerald Freilich, A denumerability formula for the rationals, Amer. Math. Monthly, Nov 1965, pp. 1013-1014.
Kevin McCrimmon, Enumeration of the positive rationals, Amer. Math. Monthly, Nov 1960, p. 868.
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.
FORMULA
If n=Product p_i^e_i, then a_n=Product p_i^f(e_i), where f(n)=n/2 if n is even and f(n)=0 if n is odd. - Reiner Martin, Jul 08 2002
a(n^2) = n, A071975(n^2) = 1, cf. A000290; a(2*(2*n-1)^2) = 2*n+1, A071975(2*(2*n-1)^2) = 2, cf. A077591. - Reinhard Zumkeller, Jul 10 2011
From Amiram Eldar, Nov 02 2023: (Start)
a(n) = sqrt(A350388(n)).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) - 1/p^(3*s-1)). (End)
EXAMPLE
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, ...
MATHEMATICA
f[{p_, a_}] := If[EvenQ[a], p^(a/2), 1]; a[n_] := Times@@(f/@FactorInteger[n])
PROG
(PARI) a(n)=local(v=factor(n)~); prod(k=1, length(v), if(v[2, k]%2, 1, v[1, k]^(v[2, k]/2)))
(Haskell)
a071974 n = product $ zipWith (^) (a027748_row n) $
map (\e -> (1 - e `mod` 2) * e `div` 2) $ a124010_row n
-- Reinhard Zumkeller, Jun 15 2012
CROSSREFS
Cf. A071975. Differs from A056622 at a(32).
For other bijective mappings from integers to positive rationals see A002487, A020652/A020653, A038568/A038569, A229994/A077610, A295515.
Sequence in context: A336643 A334039 A076933 * A056622 A331738 A306333
KEYWORD
nonn,frac,easy,nice,mult
AUTHOR
N. J. A. Sloane, Jun 19 2002
EXTENSIONS
More terms from Reiner Martin, Jul 08 2002
Additional references supplied by Kevin Ryde added by N. J. A. Sloane, May 31 2012
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)