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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
- All primes prime numbers appear in this sequence, in increasing order,
- The derived sequence b is unbounded,
Cf. A036552 (a(2n) is divisible by a(2n-1)).
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).
To compute a(2n) and a(2n+1): we take the least unused multiple of a(2n-1) with an unused proper divisor: the multiple gives a(2n) and the divisor gives a(2n+1).
The first terms, alongside their p-adic valuations with respect to p=2, 3, 5 and 7 (with 0's omitted), are:
1 1 0 0 0 0
1 1
2 4 2 0 0 0
3 2 1 0 0 0
4 6 1 1 0 0
5 3 0 1 0 0
6 15 0 1 1 0
7 5 0 0 1 0
8 20 2 0 1 0
9 10 1 0 1 0
10 40 3 0 1 0
11 8 3 0 0 0
12 24 3 1 0 0
13 12 2 1 0 0
14 36 2 2 0 0
15 9 0 2 0 0
16 54 1 3 0 0
17 18 1 2 0 0
18 90 1 2 1 0
19 30 1 1 1 0
20 120 3 1 1 0
21 60 2 1 1 0
22 180 2 2 1 0
23 45 0 2 1 0
24 135 0 3 1 0
451 524880 4 8 1 0
452 1574640 4 9 1 0
453 787320 3 9 1 0
455 7 0 0 0 1
456 28 2 0 0 1
457 14 1 0 0 1
458 42 1 1 0 1
The first 250 000 terms are 7-smooth numbersmultiple of 2 occurs at n=2: a(2)=4, and a(3)=2.
Is this a permutation The first multiple of A002473 ?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.
Conjectures:
- All primes appear in this sequence, in increasing order,
- This sequence is a permutation of the natural numbers.
Cf. A002473.
allocated Lexicographically earliest sequence of distinct terms such that, for Rémy Sigristany 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
1,2
The first 250 000 terms are 7-smooth numbers.
Is this a permutation of A002473 ?
To compute a(2n) and a(2n+1): we take the least unused multiple of a(2n-1) with an unused proper divisor: the multiple gives a(2n) and the divisor gives a(2n+1).
The first terms, alongside their valuations with respect to 2, 3, 5 and 7 are:
n a(n) v2 v3 v5 v7
--- ------- -- -- -- --
1 1 0 0 0 0
2 4 2 0 0 0
3 2 1 0 0 0
4 6 1 1 0 0
5 3 0 1 0 0
6 15 0 1 1 0
7 5 0 0 1 0
8 20 2 0 1 0
9 10 1 0 1 0
10 40 3 0 1 0
11 8 3 0 0 0
12 24 3 1 0 0
13 12 2 1 0 0
14 36 2 2 0 0
15 9 0 2 0 0
16 54 1 3 0 0
17 18 1 2 0 0
18 90 1 2 1 0
19 30 1 1 1 0
20 120 3 1 1 0
21 60 2 1 1 0
22 180 2 2 1 0
23 45 0 2 1 0
24 135 0 3 1 0
...
451 524880 4 8 1 0
452 1574640 4 9 1 0
453 787320 3 9 1 0
454 5511240 3 9 1 1
455 7 0 0 0 1
456 28 2 0 0 1
457 14 1 0 0 1
458 42 1 1 0 1
Cf. A002473.
allocated
nonn
Rémy Sigrist, Feb 04 2017
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allocated for Rémy Sigrist
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