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A071974 Numerator of rational number i/j such that Sagher map sends i/j to n. 4
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.

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]

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

David M. Bradley, Counting the Positive Rationals: A Brief Survey, arXiv:math/0509025

Gerald Freilich, A denumerability formula for the rationals, Amer. Math. Monthly, Nov 1965, p1013-1014

Kevin McCrimmon, Enumeration of the positive rationals, Amer. Math. Monthly, Nov 1960, p868.

Y. 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 (reinermartin(AT)hotmail.com), Jul 08 2002

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).

Cf. A027748, A124010.

Sequence in context: A162154 A134505 A076933 * A056622 A237983 A129265

Adjacent sequences:  A071971 A071972 A071973 * A071975 A071976 A071977

KEYWORD

nonn,frac,easy,nice,mult

AUTHOR

N. J. A. Sloane, Jun 19 2002

EXTENSIONS

More terms from Reiner Martin (reinermartin(AT)hotmail.com), 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 April 25 00:50 EDT 2014. Contains 240991 sequences.