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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. 3
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, 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, 13, 8, 12, 9, 11, 10, 10, 12, 11, 11, 13, 9, 14, 8, 16, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

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

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 one. - Thomas Ordowski, Sep 04 2014

Example: ..., 5, 13, 1, 17, 2, 13, 3, 17, 4; ...

General: ..., k,  p, 1,  q, 2,  p, 3,  q, ..., k-1; ...

- Thomas Ordowski, Sep 08 2014

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000, (first 1000 terms from T. D. Noe)

FORMULA

a(n)*gcd(a(n-1),a(n+1)) = gcd(A088178(n-1),A088178(n)). - Thomas Ordowski, Jun 29 2015

EXAMPLE

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.

MAPLE

A[1]:= 1: A[2]:= 1: S:= {1}:

for n from 3 to 100 do

  Sp:= select(type, map(s -> s/A[n-1], S), integer);

  if nops(Sp) = Sp[-1] then A[n]:= Sp[-1]+1

  else A[n]:= min({$1..Sp[-1]} minus Sp)

  fi;

  S:= S union {A[n-1]*A[n]};

od:

seq(A[n], n=1..100); # Robert Israel, Aug 28 2014

MATHEMATICA

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 *)

CROSSREFS

Cf. A088178.

Sequence in context: A304529 A071281 A204995 * A028507 A096226 A155980

Adjacent sequences:  A088174 A088175 A088176 * A088178 A088179 A088180

KEYWORD

nonn

AUTHOR

John W. Layman, Sep 22 2003

STATUS

approved

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Last modified August 16 23:58 EDT 2018. Contains 313809 sequences. (Running on oeis4.)