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.

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

Offset: 1

Author

Leroy Quet, May 06 2009

Keywords

• nonn
• look

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

Crossrefs

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

For records see A151545, A151547.

Programs

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

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
# alternative
A160256 := proc(n)
    local r,m,a,fnd,i ;
    option remember;
    if n <=2 then
        n;
    else
        r := procname(n-2)/igcd(procname(n-1),procname(n-2)) ;
        for m from 1 do
            a := m*r ;
            fnd := false;
            for i from 1 to n-1 do
                if procname(i) = a then
                    fnd := true;
                    break;
                end if;
            end do:
            if not fnd then
                return a ;
            end if;
        end do:
    end if;
end proc:
seq(A160256(n),n=1..40) ; # R. J. Mathar, Jul 30 2024

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 *)

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

Python

from math 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

Extensions

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

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