OFFSET
1,2
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.
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
EXAMPLE
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Feb 04 2017
STATUS
approved