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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). 9

%I

%S 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 27,162,81,324,108,216,72,144,16,64,32,96,48,240,80,320,160,640,128,

%U 384,192,576,288,864,432,1296,648,1944,243,972,486,1458,729,3645,405

%N 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 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 The first multiple of 2 occurs at n=2: a(2)=4, and a(3)=2.

%C The first multiple of 3 occurs at n=4: a(4)=6, and a(5)=3,

%C The first multiple of 5 occurs at n=6: a(6)=15, and a(7)=5.

%C The first multiple of 7 occurs at n=454: a(454)=5511240, and a(455)=7.

%C The first multiple of 11 occurs at n=889838: a(889838)=627667978163491186346557440000000000000, and a(889839)=11.

%C 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 n b(n) a(n)

%C ------ ------- ----

%C 2 1 2^2

%C 7 3 5

%C 19 4 2*3*5

%C 33 14 2^4

%C 73 27 5^2

%C 455 243 7

%C 1439 248 7^2

%C 3069 275 7^3

%C 10567 276 7^5

%C 41709 768 7^8

%C 85179 1169 7^10

%C 889839 >110162 11

%C Conjectures:

%C - All prime numbers appear in this sequence, in increasing order,

%C - The derived sequence b is unbounded,

%C - This sequence is a permutation of the natural numbers.

%H Rémy Sigrist, <a href="/A281978/b281978.txt">Table of n, a(n) for n = 1..25000</a>

%H Rémy Sigrist, <a href="/A281978/a281978.gp.txt">PARI program for A281978</a>

%H Rémy Sigrist, <a href="/A281978/a281978.png">Logarithmic scatterplot of the first million terms</a>

%e The first terms, alongside their p-adic valuations with respect to p=2, 3, 5 and 7 (with 0's omitted), are:

%e n a(n) v2 v3 v5 v7

%e --- ------- -- -- -- --

%e 1 1

%e 2 4 2

%e 3 2 1

%e 4 6 1 1

%e 5 3 1

%e 6 15 1 1

%e 7 5 1

%e 8 20 2 1

%e 9 10 1 1

%e 10 40 3 1

%e 11 8 3

%e 12 24 3 1

%e 13 12 2 1

%e 14 36 2 2

%e 15 9 2

%e 16 54 1 3

%e 17 18 1 2

%e 18 90 1 2 1

%e 19 30 1 1 1

%e 20 120 3 1 1

%e 21 60 2 1 1

%e 22 180 2 2 1

%e 23 45 2 1

%e 24 135 3 1

%e ...

%e 451 524880 4 8 1

%e 452 1574640 4 9 1

%e 453 787320 3 9 1

%e 454 5511240 3 9 1 1

%e 455 7 1

%e 456 28 2 1

%e 457 14 1 1

%e 458 42 1 1 1

%Y Cf. A036552 (a(2n) is divisible by a(2n-1)).

%K nonn

%O 1,2

%A _Rémy Sigrist_, Feb 04 2017

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Last modified August 4 15:33 EDT 2020. Contains 336201 sequences. (Running on oeis4.)