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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.

%I A088178 #61 Oct 21 2021 12:36:45
%S A088178 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,
%T A088178 42,35,45,27,33,44,48,60,50,70,49,56,64,72,54,66,55,65,13,17,34,26,39,
%U A088178 51,68,52,78,84,98,63,81,90,80,88,77,91,104,96,108,99,110,100,120,132
%N A088178 Sequence of distinct products b(n)*b(n+1), n=1,2,3,..., of the terms b(n) of A088177.
%C A088178 This is a permutation of the natural numbers (see the following comments).
%C A088178 Comments from _Thomas Ordowski_, Aug 24 2014 to Sep 07 2014: (Start)
%C A088178 If a(n) is a prime then a(m) > a(n) for m > n.
%C A088178 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)}.
%C A088178 The numbers in A033476 appear in increasing order.
%C A088178 It seems that the squarethe terms in s of the natural numbers also appear in increasing order, but A087811 are not strictly increasing.
%C A088178 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).
%C A088178 Theorem: the sequence a(n) is a permutation of the natural numbers. Proof: see my note to A088177.
%C A088178 At most two consecutive terms can form a decreasing subsequence.
%C A088178 (End)
%C A088178 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
%C A088178 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
%H A088178 Ivan Neretin, <a href="/A088178/b088178.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Michael De Vlieger)
%F A088178 a(n) = A088177(n)* A088177(n+1).
%F A088178 a(m) < a(n)^2 for m < n. - _Thomas Ordowski_, Sep 02 2014
%t A088178 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 *)
%o A088178 (Python)
%o A088178 from itertools import islice
%o A088178 def A088178(): # generator of terms
%o A088178     yield 1
%o A088178     p, a = {1}, 1
%o A088178     while True:
%o A088178         n, na = 1, a
%o A088178         while na in p:
%o A088178             n += 1
%o A088178             na += a
%o A088178         p.add(na)
%o A088178         a = n
%o A088178         yield na
%o A088178 A088178_list = list(islice(A088178(),20)) # _Chai Wah Wu_, Oct 21 2021
%Y A088178 Cf. A088177, A341492, A348437-A348439.
%Y A088178 Records: A348442, A348443.
%K A088178 nonn,look
%O A088178 1,2
%A A088178 _John W. Layman_, Sep 22 2003
%E A088178 Edited by _N. J. A. Sloane_, Oct 18 2021