OFFSET
1,4
COMMENTS
A023022(n) and A245677(n) give number and numerator of sum of fractions a(k)/A182973(k) such that a(k) + A182973(k) = n. - Reinhard Zumkeller, Jul 30 2014
REFERENCES
S. Cook, Problem 511: An Enumeration Problem, Journal of Recreational Mathematics, Vol. 9:2 (1976-77), 137. Solution by the Problem Editor, JRM, Vol. 10:2 (1977-78), 122-123.
R. K. Guy, Unsolved Problems in Number Theory (UPINT), Section D11.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Yiu, Recreational Mathematics, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633.
EXAMPLE
Positive fractions < 1 listed by increasing sum of numerator and denominator, and by increasing numerator for equal sums:
1/2
1/3
1/4 2/3
1/5
1/6 2/5 3/4
1/7 3/5
1/8 2/7 4/5
1/9 3/7
1/10 2/9 3/8 4/7 5/6
1/11 5/7
1/12 2/11 3/10 4/9 5/8 6/7
1/13 3/11 5/9
1/14 2/13 4/11 7/8
1/15 3/13 5/11 7/9
1/16 2/15 3/14 4/13 5/12 6/11 7/10 8/9
1/17 5/13 7/11
1/18 2/17 3/16 4/15 5/14 6/13 7/12 8/11 9/10
1/19 3/17 7/13 9/11
MAPLE
t1:=[];
for n from 2 to 40 do
t1:=[op(t1), 1/(n-1)];
for i from 2 to floor((n-1)/2) do
if gcd(i, n-i)=1 then t1:=[op(t1), i/(n-i)]; fi; od:
od:
t1;
MATHEMATICA
t1={}; For[n=2, n <= 40, n++, AppendTo[t1, 1/(n-1)]; For[i=2, i <= Floor[(n-1)/2], i++, If[GCD[i, n-i] == 1, AppendTo[t1, i/(n-i)]]]]; t1 // Numerator // Rest (* Jean-François Alcover, Jan 20 2015, translated from Maple *)
PROG
(Pascal) program a182972;
var
num, den, n: longint;
function gcd(i, j: longint):longint;
begin
repeat
if i>j then i:=i mod j else j:=j mod i;
until (i=0) or (j=0);
if i=0 then gcd:=j else gcd:=i;
end;
begin
num:=1; den:=1; n:=0;
repeat
repeat
inc(num); dec(den);
if num>=den then
begin
inc(den, num); num:=1;
end;
until gcd(num, den)=1;
inc(n); writeln(n, ' ', num);
until n=100000;
end.
(Haskell)
a182972 n = a182972_list !! (n-1)
a182972_list = map fst $ concatMap q [3..] where
q x = [(num, den) | num <- [1 .. div x 2],
let den = x - num, gcd num den == 1]
-- Reinhard Zumkeller, Jul 29 2014
(Python)
from itertools import count, islice
from math import gcd
def A182972_gen(): # generator of terms
return (i for n in count(2) for i in range(1, 1+(n-1>>1)) if gcd(i, n-i)==1)
CROSSREFS
Essentially the same as A333856.
KEYWORD
nonn,easy,frac,nice
AUTHOR
William Rex Marshall, Dec 16 2010
EXTENSIONS
Corrected by William Rex Marshall, Aug 12 2013
STATUS
approved