

A088178


Sequence of distinct products b(n)*b(n+1), n=1,2,3,..., of the terms b(n) of A088177.


2



1, 2, 4, 6, 3, 5, 10, 8, 12, 9, 15, 20, 16, 24, 18, 21, 7, 11, 22, 14, 28, 32, 40, 25, 30, 36, 42, 35, 45, 27, 33, 44, 48, 60, 50, 70, 49, 56, 64, 72, 54, 66, 55, 65, 13, 17, 34, 26, 39, 51, 68, 52, 78, 84, 98, 63, 81, 90, 80, 88, 77, 91, 104, 96, 108, 99, 110, 100, 120, 132
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OFFSET

1,2


COMMENTS

It appears that this is a permutation of the natural numbers.
If a(n) is a prime then a(m) > a(n) for m > n. Conjecture: the term a(n) is a prime if and only if every number < a(n) belongs to the set {a(1), a(2), ..., a(n1)}.  Thomas Ordowski, Aug 24 2014
The numbers A033476 appear in increasing order. It seems that squares of natural numbers also increasing, but A087811 not strictly increasing.  Thomas Ordowski, Sep 01 2014
Lemma: the sequence a(n) is a permutation of all natural numbers iff b(n) = 1 for infinitely many n, where b(n) = A088177(n), because after every b(n) = 1 is the smallest missing number in the sequence a(n). Theorem: the sequence a(n) is a permutation of the natural numbers. Proof: see my note to the A088177.  Thomas Ordowski, Sep 04 2014
At most two consecutive terms form a decreasing subsequence.  Thomas Ordowski, Sep 07 2014
Definition in other words: at step n, choose a(n) as the minimum unused multiple of the auxiliary number r, which is initialized as r=1 before the first step and changed to a(n)/r after each step.  Ivan Neretin, May 04 2015


LINKS

Michael De Vlieger and Ivan Neretin, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michael De Vlieger)


FORMULA

a(n) = A088177(n)* A088177(n+1).
a(m) < a(n)^2 for m < n.  Thomas Ordowski, Sep 02 2014


MATHEMATICA

a088177[n_Integer] := Module[{t = {1, 1}}, Do[AppendTo[t, 1]; While[Length[Union[Most[t]*Rest[t]]] < i  1, t[[1]]++], {i, 3, n}]; t]; a088178[n_Integer] := Last[a088177[n]]*Last[a088177[n + 1]]; a088178 /@ Range[120] (* Michael De Vlieger, Aug 30 2014, based on T. D. Noe's script at A088177 *)


CROSSREFS

Cf. A088177.
Sequence in context: A175213 A104492 A075075 * A259840 A161184 A140645
Adjacent sequences: A088175 A088176 A088177 * A088179 A088180 A088181


KEYWORD

nonn,look


AUTHOR

John W. Layman, Sep 22 2003


STATUS

approved



