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).

1, 4, 2, 6, 3, 15, 5, 20, 10, 40, 8, 24, 12, 36, 9, 54, 18, 90, 30, 120, 60, 180, 45, 135, 27, 162, 81, 324, 108, 216, 72, 144, 16, 64, 32, 96, 48, 240, 80, 320, 160, 640, 128, 384, 192, 576, 288, 864, 432, 1296, 648, 1944, 243, 972, 486, 1458, 729, 3645, 405

Offset: 1

Author

Rémy Sigrist, Feb 04 2017

Keywords

• nonn

Comments

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).

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

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

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

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

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

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:

n b(n) a(n)

------ ------- ----

2 1 2^2

7 3 5

19 4 2*3*5

33 14 2^4

73 27 5^2

455 243 7

1439 248 7^2

3069 275 7^3

10567 276 7^5

41709 768 7^8

85179 1169 7^10

889839 >110162 11

Conjectures:

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

- The derived sequence b is unbounded,

- This sequence is a permutation of the natural numbers.

Examples

			The first terms, alongside their p-adic valuations with respect to p=2, 3, 5 and 7 (with 0's omitted), are:
n       a(n)  v2  v3  v5  v7
---  -------  --  --  --  --
1          1
2          4   2
3          2   1
4          6   1   1
5          3       1
6         15       1   1
7          5           1
8         20   2       1
9         10   1       1
10        40   3       1
11         8   3
12        24   3   1
13        12   2   1
14        36   2   2
15         9       2
16        54   1   3
17        18   1   2
18        90   1   2   1
19        30   1   1   1
20       120   3   1   1
21        60   2   1   1
22       180   2   2   1
23        45       2   1
24       135       3   1
...
451   524880   4   8   1
452  1574640   4   9   1
453   787320   3   9   1
454  5511240   3   9   1   1
455        7               1
456       28   2           1
457       14   1           1
458       42   1   1       1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..25000
• Rémy Sigrist, PARI program for A281978
• Rémy Sigrist, Logarithmic scatterplot of the first million terms

Crossrefs

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