A182972 - OEIS

%I #47 Aug 26 2024 11:42:13

%S 1,1,1,2,1,1,2,3,1,3,1,2,4,1,3,1,2,3,4,5,1,5,1,2,3,4,5,6,1,3,5,1,2,4,

%T 7,1,3,5,7,1,2,3,4,5,6,7,8,1,5,7,1,2,3,4,5,6,7,8,9,1,3,7,9,1,2,4,5,8,

%U 10,1,3,5,7,9,1,2,3,4,5,6,7,8,9,10,11,1,5,7,11,1,2,3,4,6,7,8,9,11,12,1,3,5,7,9,11

%N Numerators of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator.

%C 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

%D 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.

%D R. K. Guy, Unsolved Problems in Number Theory (UPINT), Section D11.

%H Reinhard Zumkeller, <a href="/A182972/b182972.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Yiu, <a href="https://web.archive.org/web/20220720212753/http://math.fau.edu/Yiu/RecreationalMathematics2003.pdf">Recreational Mathematics</a>, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633.

%H <a href="/index/Ra#rational">Index entries for sequences related to enumerating the rationals</a>

%e Positive fractions < 1 listed by increasing sum of numerator and denominator, and by increasing numerator for equal sums:

%e 1/2

%e 1/3

%e 1/4 2/3

%e 1/5

%e 1/6 2/5 3/4

%e 1/7 3/5

%e 1/8 2/7 4/5

%e 1/9 3/7

%e 1/10 2/9 3/8 4/7 5/6

%e 1/11 5/7

%e 1/12 2/11 3/10 4/9 5/8 6/7

%e 1/13 3/11 5/9

%e 1/14 2/13 4/11 7/8

%e 1/15 3/13 5/11 7/9

%e 1/16 2/15 3/14 4/13 5/12 6/11 7/10 8/9

%e 1/17 5/13 7/11

%e 1/18 2/17 3/16 4/15 5/14 6/13 7/12 8/11 9/10

%e 1/19 3/17 7/13 9/11

%e (this is A182972/A182973).

%p t1:=[];

%p for n from 2 to 40 do

%p t1:=[op(t1),1/(n-1)];

%p for i from 2 to floor((n-1)/2) do

%p    if gcd(i,n-i)=1 then t1:=[op(t1),i/(n-i)]; fi; od:

%p od:

%p t1;

%t 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 *)

%o (Pascal) program a182972;

%o var

%o   num,den,n: longint;

%o function gcd(i,j: longint):longint;

%o begin

%o   repeat

%o     if i>j then i:=i mod j else j:=j mod i;

%o   until (i=0) or (j=0);

%o   if i=0 then gcd:=j else gcd:=i;

%o end;

%o begin

%o   num:=1; den:=1; n:=0;

%o   repeat

%o     repeat

%o       inc(num); dec(den);

%o       if num>=den then

%o       begin

%o         inc(den,num); num:=1;

%o       end;

%o     until gcd(num,den)=1;

%o     inc(n); writeln(n,' ',num);

%o   until n=100000;

%o end.

%o (Haskell)

%o a182972 n = a182972_list !! (n-1)

%o a182972_list = map fst $ concatMap q [3..] where

%o    q x = [(num, den) | num <- [1 .. div x 2],

%o                        let den = x - num, gcd num den == 1]

%o -- _Reinhard Zumkeller_, Jul 29 2014

%o (Python)

%o from itertools import count, islice

%o from math import gcd

%o def A182972_gen(): # generator of terms

%o     return (i for n in count(2) for i in range(1,1+(n-1>>1)) if gcd(i,n-i)==1)

%o A182972_list = list(islice(A182972_gen(),10)) # _Chai Wah Wu_, Aug 28 2023

%Y Cf. A182973 (denominators), A366191 (interleaved).

%Y Cf. A020652, A038567, A182974, A182975, A182976.

%Y Cf. A023022, A245677, A245678, A245718.

%Y Essentially the same as A333856.

%K nonn,easy,frac,nice

%O 1,4

%A _William Rex Marshall_, Dec 16 2010

%E Corrected by _William Rex Marshall_, Aug 12 2013