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A281978 Lexicographically earliest sequence of distinct terms such that, for any n>0, a(2n) is divisible by a(2n-1) and by a(2n+1).

%I A281978 #10 Feb 07 2017 04:04:38
%S A281978 1,4,2,6,3,15,5,20,10,40,8,24,12,36,9,54,18,90,30,120,60,180,45,135,
%T A281978 27,162,81,324,108,216,72,144,16,64,32,96,48,240,80,320,160,640,128,
%U A281978 384,192,576,288,864,432,1296,648,1944,243,972,486,1458,729,3645,405
%N A281978 Lexicographically earliest sequence of distinct terms such that, for any n>0, a(2n) is divisible by a(2n-1) and by a(2n+1).
%C A281978 To compute a(2n) and a(2n+1): we take the least unseen multiple of a(2n-1) with an unseen proper divisor: the multiple gives a(2n) and the least proger divisor gives a(2n+1).
%C A281978 The first multiple of 2 occurs at n=2: a(2)=4, and a(3)=2.
%C A281978 The first multiple of 3 occurs at n=4: a(4)=6, and a(5)=3,
%C A281978 The first multiple of 5 occurs at n=6: a(6)=15, and a(7)=5.
%C A281978 The first multiple of 7 occurs at n=454: a(454)=5511240, and a(455)=7.
%C A281978 The first multiple of 11 occurs at n=889838: a(889838)=627667978163491186346557440000000000000, and a(889839)=11.
%C A281978 For n>1, let b(n)=least k>0 such that a(n+k)<>a(n)*a(k+1); the first records for b are:
%C A281978 n       b(n)     a(n)
%C A281978 ------  -------  ----
%C A281978 2             1  2^2
%C A281978 7             3  5
%C A281978 19            4  2*3*5
%C A281978 33           14  2^4
%C A281978 73           27  5^2
%C A281978 455         243  7
%C A281978 1439        248  7^2
%C A281978 3069        275  7^3
%C A281978 10567       276  7^5
%C A281978 41709       768  7^8
%C A281978 85179      1169  7^10
%C A281978 889839  >110162  11
%C A281978 Conjectures:
%C A281978 - All prime numbers appear in this sequence, in increasing order,
%C A281978 - The derived sequence b is unbounded,
%C A281978 - This sequence is a permutation of the natural numbers.
%H A281978 Rémy Sigrist, <a href="/A281978/b281978.txt">Table of n, a(n) for n = 1..25000</a>
%H A281978 Rémy Sigrist, <a href="/A281978/a281978.gp.txt">PARI program for A281978</a>
%H A281978 Rémy Sigrist, <a href="/A281978/a281978.png">Logarithmic scatterplot of the first million terms</a>
%e A281978 The first terms, alongside their p-adic valuations with respect to p=2, 3, 5 and 7 (with 0's omitted), are:
%e A281978 n       a(n)  v2  v3  v5  v7
%e A281978 ---  -------  --  --  --  --
%e A281978 1          1
%e A281978 2          4   2
%e A281978 3          2   1
%e A281978 4          6   1   1
%e A281978 5          3       1
%e A281978 6         15       1   1
%e A281978 7          5           1
%e A281978 8         20   2       1
%e A281978 9         10   1       1
%e A281978 10        40   3       1
%e A281978 11         8   3
%e A281978 12        24   3   1
%e A281978 13        12   2   1
%e A281978 14        36   2   2
%e A281978 15         9       2
%e A281978 16        54   1   3
%e A281978 17        18   1   2
%e A281978 18        90   1   2   1
%e A281978 19        30   1   1   1
%e A281978 20       120   3   1   1
%e A281978 21        60   2   1   1
%e A281978 22       180   2   2   1
%e A281978 23        45       2   1
%e A281978 24       135       3   1
%e A281978 ...
%e A281978 451   524880   4   8   1
%e A281978 452  1574640   4   9   1
%e A281978 453   787320   3   9   1
%e A281978 454  5511240   3   9   1   1
%e A281978 455        7               1
%e A281978 456       28   2           1
%e A281978 457       14   1           1
%e A281978 458       42   1   1       1
%Y A281978 Cf. A036552 (a(2n) is divisible by a(2n-1)).
%K A281978 nonn
%O A281978 1,2
%A A281978 _Rémy Sigrist_, Feb 04 2017