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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 (list; graph; refs; listen; history; text; internal format)
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(n-1)}. - 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

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Last modified January 17 16:47 EST 2018. Contains 297822 sequences.