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.

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

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 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]

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 *)

Prog

(Python)
from itertools import islice
def A088177(): # generator of terms
    yield 1
    yield 1
    p, a = {1}, 1
    while True:
        n = 1
        while n*a in p:
            n += 1
        p.add(n*a)
        a = n
        yield n
A088177_list = list(islice(A088177(), 20)) # Chai Wah Wu, Oct 21 2021

Crossrefs

Cf. A088178, A307720, A307730, A341492, A348437, A348438, A348439.

Records: A348440, A348441.

Sequence in context: A368400 A325099 A370777 * A335917 A028507 A096226

Adjacent sequences: A088174 A088175 A088176 * A088178 A088179 A088180

Keyword

nonn,nice

Author

John W. Layman, Sep 22 2003

Status

approved