A088178 Sequence of distinct products b(n)*b(n+1), n=1,2,3,..., of the terms b(n) of A088177.

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

Offset

1,2

Comments

This is a permutation of the natural numbers (see the following comments).

Comments from Thomas Ordowski, Aug 24 2014 to Sep 07 2014: (Start)

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)}.

The numbers in A033476 appear in increasing order.

It seems that the squarethe terms in s of the natural numbers also appear in increasing order, but A087811 are not strictly increasing.

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 A088177.

At most two consecutive terms can form a decreasing subsequence.

(End)

An equivalent definition. At step n, choose a(n) to be the smallest unused multiple of the auxiliary number r, which is initially 1 and is changed to a(n)/r after each step. - Ivan Neretin, May 04 2015

Considered as a permutation of the positive integers, there are finite cycles (1), (2), (3, 4, 6, 5), (8), (11, 18, 15), (52), and probably others. The cycle containing 7, on the other hand, is ( ..., 85, 46, 17, 7, 10, 9, 12, 20, 14, 24, 25, 30, 27, 42, 66, 99, 160, 308, 343, 430, 517, 902, ... ), and may be infinite. The inverse permutation is A341492. - N. J. A. Sloane, Oct 19 2021

Links

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 *)

Prog

(Python)
from itertools import islice
def A088178(): # generator of terms
    yield 1
    p, a = {1}, 1
    while True:
        n, na = 1, a
        while na in p:
            n += 1
            na += a
        p.add(na)
        a = n
        yield na
A088178_list = list(islice(A088178(), 20)) # Chai Wah Wu, Oct 21 2021

Crossrefs

Cf. A088177, A341492, A348437-A348439.

Records: A348442, A348443.

Sequence in context: A104492 A331522 A075075 * A259840 A161184 A349543

Adjacent sequences: A088175 A088176 A088177 * A088179 A088180 A088181

Keyword

nonn,look

Author

John W. Layman, Sep 22 2003

Extensions

Edited by N. J. A. Sloane, Oct 18 2021

Status

approved