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

1,2

Comments

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

Links

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

Example

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.

Prog

(PARI) copywo(v, k)=vector(#v-1, i, v[if(i<k, i, i+1)])
rprat(n)=local(v, i); v=vector(eulerphi(n)); i=0; for(k=1, n, if(gcd(k, n)==1, v[i++ ]=k/(n-k))); v
invecn(v, n, x)=for(k=1, n, if(v[k]==x, return(1))); 0
al(n)={local(v, pend, last, k, try);
v=vector(n); v[1]=1; pend=[]; last=2;
for(i=2, n,
k=1; while(1,
if(k>#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}
(Haskell)
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

Crossrefs

This ordering of the rationals is A038566/A020653.

Cf. A002487.

Sequence in context: A304752 A113552 A282291 * A064736 A303751 A304531

Adjacent sequences: A176349 A176350 A176351 * A176353 A176354 A176355

Keyword

nice,nonn

Author

Franklin T. Adams-Watters, Apr 15 2010

Extensions

Definition stated more precisely by Reinhard Zumkeller, Oct 30 2012

Status

approved