A218535 -id:A218535 - cp's OEIS frontend

A176352 Order the positive rationals by numerator+denominator, then by numerator. a(n+1) = a(n)*r, where r is the first unused positive rational that makes a(n+1) an integer not already in the sequence.

1, 2, 6, 3, 12, 4, 20, 5, 30, 45, 9, 15, 10, 25, 175, 70, 42, 7, 56, 8, 28, 21, 49, 14, 126, 168, 210, 90, 72, 16, 160, 60, 50, 225, 270, 27, 297, 33, 88, 11, 132, 231, 165, 264, 24, 54, 63, 36, 120, 75, 105, 189, 84, 462, 396, 108, 1404, 117, 65, 910, 273, 1001, 182, 13

Offset: 1

Franklin T. Adams-Watters, Apr 15 2010

It appears that this sequence is a permutation of the positive integers.
It appears that every positive rational except 1 occurs as the ratio of consecutive terms.
A218454 gives smallest numbers m such that a(m)=n; a(A176352(n))=n. - Reinhard Zumkeller, Oct 30 2012
A218535(n) = gcd(a(n),a(n+1)); A218533(n)/A218534(n) = a(n)/a(n+1). - Reinhard Zumkeller, Nov 10 2012

			After a(6)=4, we have used ratios 1/2, 2, 1/3, and 3. 1/4 would give 1, which is already used. 2/3 would give 8/3, not an integer; 3/2 would give 6, already used; and ratio 4 is already used. 1/5 would not produce an integer; next is 5, giving a(7) = 4*5 = 20.

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

This ordering of the rationals is A038566/A020653.

Cf. A002487.

import Data.Ratio ((%), numerator, denominator)
import Data.List (delete)
import Data.Set (singleton, insert, member)
a176352 n = a176352_list !! (n-1)
a176352_list = 1 : f 1 (singleton 1) (concat $ drop 2 $
   zipWith (zipWith (%)) a038566_tabf $ map reverse a038566_tabf)
   where f x ws qs = h qs
           where h (r:rs) | denominator y /= 1 || v `member` ws = h rs
                          | otherwise = v : f y (insert v ws) (delete r qs)
                          where v = numerator y; y = x * r
-- Reinhard Zumkeller, Oct 30 2012

copywo(v,k)=vector(#v-1,i,v[if(i#pend,pend=concat(pend,rprat(last++)));
try=v[i-1]*pend[k];
if(denominator(try)==1&!invecn(v,i-1,try),
pend=copywo(pend,k);v[i]=try;break);
k++));v}

Definition stated more precisely by Reinhard Zumkeller, Oct 30 2012

A218533 Numerators of the ratios of consecutive terms in A176352.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 2, 5, 3, 3, 2, 1, 5, 5, 6, 1, 7, 2, 4, 3, 7, 1, 3, 4, 7, 5, 9, 1, 8, 6, 2, 5, 10, 1, 9, 3, 8, 1, 4, 7, 5, 11, 4, 6, 7, 3, 8, 5, 5, 9, 2, 7, 11, 1, 12, 9, 1, 10, 3, 11, 14, 1, 5, 13, 3, 13, 2, 11, 4, 13, 1, 8, 7, 13, 4, 11, 9, 7, 16, 2

Offset: 1

Reinhard Zumkeller, Nov 10 2012

a(n) = numerator of A176352(n) / A176352(n+1).

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

Cf. A218533 (denominator).

import Data.Ratio ((%), numerator)
a218533 n = a176352 n `div` a218535 n
a218533_list = map numerator $ zipWith (%) a176352_list $ tail a176352_list

a(n) = A176352(n) / A218535(n).

A218534 Denominators of the ratios of consecutive terms in A176352.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 10 2012

a(n) = denominator of A176352(n) / A176352(n+1).

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

Cf. A218533 (numerator).

import Data.Ratio ((%), denominator)
a218534 n = a176352 (n + 1) `div` a218535 n
a218534_list = map denominator $ zipWith (%) a176352_list $ tail a176352_list

a(n) = A176352(n+1) / A218535(n).

cp's OEIS Frontend

A176352 Order the positive rationals by numerator+denominator, then by numerator. a(n+1) = a(n)*r, where r is the first unused positive rational that makes a(n+1) an integer not already in the sequence.

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A218533 Numerators of the ratios of consecutive terms in A176352.

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A218534 Denominators of the ratios of consecutive terms in A176352.

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