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A160256 a(1)=1, a(2)=2. For n >=3, a(n) = the smallest positive integer not occurring earlier in the sequence such that a(n)*a(n-1)/a(n-2) is an integer. 8
1, 2, 3, 4, 6, 8, 9, 16, 18, 24, 12, 10, 30, 5, 36, 15, 48, 20, 60, 7, 120, 14, 180, 21, 240, 28, 300, 35, 360, 42, 420, 11, 840, 22, 1260, 33, 1680, 44, 2100, 55, 2520, 66, 2940, 77, 3360, 88, 3780, 110, 378, 165, 126, 220, 63, 440, 189, 880, 567, 1760 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Is this sequence a permutation of the positive integers?

a(n+2)*a(n+1)/a(n) = A160257(n).

From Alois P. Heinz, May 07 2009: (Start)

After computing about 10^7 elements of A160256 we have

a(10000000) = 2099597439752627193722111679586865799879114417

a(10000001) = 992131130100042530286371815859160

Largest element so far:

a(8968546) = 24941014474345046106920043019655502800839523254002490663461\

524119982890708516899294655028121578883343551450916846444559467340663409\

549447588184641816

Still missing:

19, 23, 27, 29, 31, 32, 37, 38, 41, 43, 45, 46, 47, 53, 54, 57, 58, 59,

61, 62, 64, 67, 69, 71, 72, 73, 74, 76, 79, 81, 82, 83, 86, 87, 89, 90,

92, 93, 94, 95, 96, 97, 101, 103, 105, 106, 107, 108, 109, 111, 112, 113,

114, 115, 116, 118, 122, 123, 124, 125, 127, 128, 129, 131, 133, 134, ...

Primes in sequence so far:

2, 3, 5, 7, 11, 13, 17

The sequence consists of two subsequences, even (=red) and odd (=blue), see plot. (End)

a(n) is the least multiple of a(n-2)/gcd(a(n-1),a(n-2)) that has not previously occurred. - Thomas Ordowski, Jul 15 2015

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..130000

Alois P. Heinz, Color plot of first 600 terms

MAPLE

b:= proc(n) option remember; false end:

a:= proc(n) option remember; local k, m;

      if n<3 then b(n):=true; n

    else m:= denom(a(n-1)/a(n-2));

         for k from m by m while b(k) do od;

         b(k):= true; k

      fi

    end:

seq(a(n), n=1..100); # Alois P. Heinz, May 16 2009

MATHEMATICA

f[s_List] := Block[{k = 1, m = Denominator[ s[[ -1]]/s[[ -2]]]}, While[ MemberQ[s, k*m] || Mod[k*m*s[[ -1]], s[[ -2]]] != 0, k++ ]; Append[s, k*m]]; Nest[f, {1, 2}, 56] (* Robert G. Wilson v, May 17 2009 *)

PROG

(PARI)

LQ(nMax)={my(a1=1, a2=1, L=1/*least unseen number*/, S=[]/*used numbers above L*/);

while(1, /*cleanup*/ while( setsearch(S, L), S=setminus(S, Set(L)); L++);

/*search*/ for(a=L, nMax, a*a2%a1 & next; setsearch(S, a) & next;

print1(a", "); a1=a2; S=setunion(S, Set(a2=a)); next(2)); return(L))} \\ M. F. Hasler, May 06 2009

(PARI) L=10^4; a=vector(L); b=[1, 2]; a[1]=1; a[2]=2; sb=2; P2=2; pending=[]; sp=0; for(n=3, L, if(issquare(n), b=vecsort(concat(b, pending)); sb=n-1; while(sb>=2*P2, P2*=2); sp=0; pending=[]); c=a[n-2]/gcd(a[n-2], a[n-1]); u=0; while(1, u+=c; found=0; s=0; pow2=P2; while(pow2, s2=s+pow2; if((s2<=sb)&&(b[s2]<=u), s=s2); pow2\=2); if((s>0)&&(b[s]==u), found=1, for(i=1, sp, if(pending[i]==u, found=1; break))); if(found==0, break)); a[n]=u; pending=concat(pending, u); sp++); a \\ Robert Gerbicz, May 16 2009]

(Haskell)

import Data.List (delete)

a160256 n = a160256_list !! (n-1)

a160256_list = 1 : 2 : f 1 2 [3..] where

   f u v ws = g ws where

     g (x:xs) | mod (x * v) u == 0 = x : f v x (delete x ws)

              | otherwise          = g xs

-- Reinhard Zumkeller, Jan 31 2014

(Python)

from __future__ import division

from fractions import gcd

A160256_list, l1, l2, m, b = [1, 2], 2, 1, 1, {1, 2}

for _ in range(10**3):

....i = m

....while True:

........if not i in b:

............A160256_list.append(i)

............l1, l2, m = i, l1, l1//gcd(l1, i)

............b.add(i)

............break

........i += m # Chai Wah Wu, Dec 09 2014

CROSSREFS

Cf. A075075, A160257, A151413, A160218, A151546, A064413.

For records see A151545, A151547.

Sequence in context: A018338 A018271 A073667 * A151545 A097274 A254438

Adjacent sequences:  A160253 A160254 A160255 * A160257 A160258 A160259

KEYWORD

nonn,look

AUTHOR

Leroy Quet, May 06 2009

EXTENSIONS

More terms from M. F. Hasler, May 06 2009

Edited by N. J. A. Sloane, May 16 2009

STATUS

approved

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Last modified March 30 18:30 EDT 2017. Contains 284302 sequences.