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
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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, Jul 26 2024: (Start)
a(n) = sqrt(A350388(n)) (square root of largest unitary divisor of n that is a square).
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])
Table[Sqrt@ SelectFirst[Reverse@ Divisors@ n, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 104}] (* Michael De Vlieger, Dec 06 2018 *)
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
(Python)
from math import prod
from sympy import factorint
def A071974(n): return prod(p**(e>>1) for p, e in factorint(n).items() if e&1^1) # Chai Wah Wu, Jul 27 2024
CROSSREFS
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