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A088177 a(1)=1, a(2)=1; for n>2, a(n) is the smallest positive integer such that the products a(i)*a(i+1), i=1..n-1, are all distinct.

%I A088177 #73 Oct 21 2021 10:52:14
%S A088177 1,1,2,2,3,1,5,2,4,3,3,5,4,4,6,3,7,1,11,2,7,4,8,5,5,6,6,7,5,9,3,11,4,
%T A088177 12,5,10,7,7,8,8,9,6,11,5,13,1,17,2,13,3,17,4,13,6,14,7,9,9,10,8,11,7,
%U A088177 13,8,12,9,11,10,10,12,11,11,13,9,14,8,16,9
%N A088177 a(1)=1, a(2)=1; for n>2, a(n) is the smallest positive integer such that the products a(i)*a(i+1), i=1..n-1, are all distinct.
%C A088177 A088178 is the sequence of distinct products a(i)a(i+1), i=1,2,3,... and appears to be a permutation of the natural numbers.
%C A088177 It appears that for k>2 the k-th occurrence of 1 lies between the first occurrences of primes p(2*k-4) and p(2*k-3).  For instance, the 5th occurrence of 1 lies between the first occurrences of 13 and 17, the 6th and 7th primes, respectively. - _John W. Layman_, Nov 16 2011
%C A088177 Note that a(n) = 1 for infinitely many n, because the sequence a(n) is not bounded and beside every new prime number must be the number 1. - _Thomas Ordowski_, Sep 04 2014. [This seems a rather sketchy argument, but I have a more complete proof using arguments similar to those we used in A098550. - _N. J. A. Sloane_, Oct 18 2021]
%C A088177 Example: ..., 5, 13, 1, 17, 2, 13, 3, 17, 4; ...
%C A088177 General: ..., k,  p, 1,  q, 2,  p, 3,  q, ..., k-1; ...
%C A088177 - _Thomas Ordowski_, Sep 08 2014
%H A088177 Michael De Vlieger, <a href="/A088177/b088177.txt">Table of n, a(n) for n = 1..10000</a>, (first 1000 terms from T. D. Noe)
%F A088177 a(n)*gcd(a(n-1),a(n+1)) = gcd(A088178(n-1),A088178(n)). - _Thomas Ordowski_, Jun 29 2015
%e A088177 Given that the sequence begins 1,1,2,2,... then a(5)=3, since either of the choices a(5)=1 or a(5)=2 would lead to a repetition of one of the previous products 1,2,4 of adjacent pairs of terms.
%p A088177 A[1]:= 1: A[2]:= 1: S:= {1}:
%p A088177 for n from 3 to 100 do
%p A088177   Sp:= select(type,map(s -> s/A[n-1],S),integer);
%p A088177   if nops(Sp) = Sp[-1] then A[n]:= Sp[-1]+1
%p A088177   else A[n]:= min({$1..Sp[-1]} minus Sp)
%p A088177   fi;
%p A088177   S:= S union {A[n-1]*A[n]};
%p A088177 od:
%p A088177 seq(A[n],n=1..100); # _Robert Israel_, Aug 28 2014
%t A088177 t = {1, 1}; Do[AppendTo[t, 1]; While[Length[Union[Most[t]*Rest[t]]] < n - 1, t[[-1]]++], {n, 3, 100}]; t (* _T. D. Noe_, Nov 16 2011 *)
%o A088177 (Python)
%o A088177 from itertools import islice
%o A088177 def A088177(): # generator of terms
%o A088177     yield 1
%o A088177     yield 1
%o A088177     p, a = {1}, 1
%o A088177     while True:
%o A088177         n = 1
%o A088177         while n*a in p:
%o A088177             n += 1
%o A088177         p.add(n*a)
%o A088177         a = n
%o A088177         yield n
%o A088177 A088177_list = list(islice(A088177(),20)) # _Chai Wah Wu_, Oct 21 2021
%Y A088177 Cf. A088178, A307720, A307730, A341492, A348437, A348438, A348439.
%Y A088177 Records: A348440, A348441.
%K A088177 nonn,nice
%O A088177 1,3
%A A088177 _John W. Layman_, Sep 22 2003