A071975 Denominator of rational number i/j such that Sagher map sends i/j to n.

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, 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, 18, 55, 28, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 4, 73, 74, 3, 19, 77

Offset: 1

Author

N. J. A. Sloane, Jun 19 2002

Keywords

• nonn
• frac
• easy
• nice
• mult

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.

Examples

			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, ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.
• Index entries for sequences related to enumerating the rationals

Crossrefs

Cf. A000290, A071974.

Cf. A002117, A027748, A124010.

Programs

Haskell

a071975 n = product $ zipWith (^) (a027748_row n) $
   map (\e -> (e `mod` 2) * (e + 1) `div` 2) $ a124010_row n
-- Reinhard Zumkeller, Jun 15 2012

Mathematica

f[{p_, a_}] := If[OddQ[a], p^((a+1)/2), 1]; a[n_] := Times@@(f/@FactorInteger[n])

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))

Formula

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

From Reinhard Zumkeller, Jul 10 2011: (Start)

a(n^2) = 1, A071974(n^2) = n, cf. A000290.

a(2*(2*n-1)^2) = 2, A071974(2*(2*n-1)^2) = 2*n+1, cf. A077591.

a(2*(2*n-1)^2) = 2, A071974(2*(2*n-1)^2) = 2*n+1, cf. A077591. (End)

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

Extensions

More terms from Reiner Martin, Jul 08 2002