A088178 - OEIS

%I #61 Oct 21 2021 12:36:45

%S 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 42,35,45,27,33,44,48,60,50,70,49,56,64,72,54,66,55,65,13,17,34,26,39,

%U 51,68,52,78,84,98,63,81,90,80,88,77,91,104,96,108,99,110,100,120,132

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

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

%C Comments from _Thomas Ordowski_, Aug 24 2014 to Sep 07 2014: (Start)

%C If a(n) is a prime then a(m) > a(n) for m > n.

%C 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 The numbers in A033476 appear in increasing order.

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

%C 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 Theorem: the sequence a(n) is a permutation of the natural numbers. Proof: see my note to A088177.

%C At most two consecutive terms can form a decreasing subsequence.

%C (End)

%C 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 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 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 a(n) = A088177(n)* A088177(n+1).

%F a(m) < a(n)^2 for m < n. - _Thomas Ordowski_, Sep 02 2014

%t 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 (Python)

%o from itertools import islice

%o def A088178(): # generator of terms

%o     yield 1

%o     p, a = {1}, 1

%o     while True:

%o         n, na = 1, a

%o         while na in p:

%o             n += 1

%o             na += a

%o         p.add(na)

%o         a = n

%o         yield na

%o A088178_list = list(islice(A088178(),20)) # _Chai Wah Wu_, Oct 21 2021

%Y Cf. A088177, A341492, A348437-A348439.

%Y Records: A348442, A348443.

%K nonn,look

%O 1,2

%A _John W. Layman_, Sep 22 2003

%E Edited by _N. J. A. Sloane_, Oct 18 2021