Search: a285296 -id:a285296
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1, 3, 5, 2, 7, 4, 9, 6, 8, 11, 14, 10, 16, 12, 18, 13, 21, 15, 23, 17, 19, 25, 28, 20, 22, 26, 24, 27, 30, 32, 36, 29, 33, 38, 41, 31, 43, 39, 34, 35, 45, 40, 47, 37, 42, 49, 51, 44, 46, 48, 53, 50, 55, 52, 57, 54, 59, 61, 65, 56, 67, 62, 58, 60, 69, 63, 73
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OFFSET
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1,2
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LINKS
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EXAMPLE
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A285299
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Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^3 for some prime p.
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+10
2
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1, 8, 2, 4, 6, 9, 3, 16, 5, 24, 7, 27, 10, 12, 14, 20, 18, 15, 25, 30, 28, 22, 32, 11, 40, 13, 48, 17, 54, 19, 56, 21, 36, 26, 44, 34, 52, 38, 60, 42, 45, 33, 63, 39, 64, 23, 72, 29, 80, 31, 81, 35, 49, 70, 50, 55, 75, 65, 88, 37, 96, 41, 104, 43, 108, 46, 68
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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This sequence is a permutation of the natural numbers.
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LINKS
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EXAMPLE
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The first terms, alongside the primes p such that p^3 divides a(n)*a(n+1), are:
n a(n) p
-- ---- -
1 1 2
2 8 2
3 2 2
4 4 2
5 6 3
6 9 3
7 3 2
8 16 2
9 5 2
10 24 2
11 7 3
12 27 3
13 10 2
14 12 2
15 14 2
16 20 2
17 18 3
18 15 5
19 25 5
20 30 2
...
64 43 3
65 108 2, 3
66 46 2
...
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A285386
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Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^4 for some prime p.
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+10
2
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1, 16, 2, 8, 4, 12, 20, 24, 6, 27, 3, 32, 5, 48, 7, 64, 9, 18, 36, 28, 40, 10, 56, 14, 72, 22, 80, 11, 81, 13, 96, 15, 54, 21, 108, 30, 88, 26, 104, 34, 112, 17, 128, 19, 144, 23, 160, 25, 50, 75, 100, 44, 52, 60, 68, 76, 84, 92, 116, 120, 38, 136, 42, 135, 33
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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This sequence is a permutation of the natural numbers.
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LINKS
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EXAMPLE
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The first terms, alongside the primes p such that p^4 divides a(n)*a(n+1), are:
n a(n) p
-- ---- -
1 1 2
2 16 2
3 2 2
4 8 2
5 4 2
6 12 2
7 20 2
8 24 2
9 6 3
10 27 3
11 3 2
12 32 2
13 5 2
14 48 2
15 7 2
16 64 2
17 9 3
18 18 3
19 36 2
20 28 2
...
165 95 2
166 432 2, 3
167 87 3
...
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A285417
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Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^5 for some prime p.
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+10
2
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1, 32, 2, 16, 4, 8, 12, 24, 20, 40, 28, 48, 6, 64, 3, 81, 9, 27, 18, 54, 36, 56, 44, 72, 52, 80, 10, 96, 5, 128, 7, 160, 11, 192, 13, 224, 14, 112, 22, 144, 26, 176, 30, 162, 15, 243, 17, 256, 19, 288, 21, 320, 23, 352, 25, 125, 50, 208, 34, 240, 38, 272, 42
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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This sequence is a permutation of the natural numbers.
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LINKS
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EXAMPLE
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The first terms, alongside the primes p such that p^5 divides a(n)*a(n+1), are:
n a(n) p
-- ---- -
1 1 2
2 32 2
3 2 2
4 16 2
5 4 2
6 8 2
7 12 2
8 24 2
9 20 2
10 40 2
11 28 2
12 48 2
13 6 2
14 64 2
15 3 3
16 81 3
17 9 3
18 27 3
19 18 3
20 54 3
...
1476 7744 2
1477 811 2, 3
1478 7776 2, 3
1479 813 3
...
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A285575
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Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^2 for at least two distinct primes p.
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+10
2
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1, 36, 2, 18, 4, 9, 8, 25, 12, 3, 24, 6, 30, 10, 20, 5, 40, 15, 45, 16, 27, 28, 7, 56, 14, 42, 21, 48, 33, 44, 11, 72, 13, 52, 26, 50, 22, 54, 32, 49, 60, 35, 63, 64, 75, 39, 78, 66, 84, 51, 68, 17, 100, 19, 76, 38, 90, 34, 98, 46, 92, 23, 108, 29, 116, 58
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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The sequence can always be extended with a multiple of 36; after a multiple of 36, we can extend the sequence with the least unused number; as there are infinitely many multiples of 36, this sequence is a permutation of the natural numbers (with inverse A285576).
For any k>=0, let c_k be the lexicographically earliest sequence of distinct terms such that the product of two consecutive terms is divisible by p^2 for at least k distinct primes p; in particular we have:
- c_0 = A000027 (the natural numbers),
- c_2 = a (this sequence).
For any k>=0, c_k is a permutation of the natural numbers.
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LINKS
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EXAMPLE
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The first terms, alongside the primes p such that p^2 divides a(n)*a(n+1), are:
n a(n) p
-- ---- ----
1 1 2, 3
2 36 2, 3
3 2 2, 3
4 18 2, 3
5 4 2, 3
6 9 2, 3
7 8 2, 5
8 25 2, 5
9 12 2, 3
10 3 2, 3
11 24 2, 3
12 6 2, 3
13 30 2, 5
14 10 2, 5
15 20 2, 5
16 5 2, 5
17 40 2, 5
18 15 3, 5
19 45 2, 3
20 16 2, 3
...
115 160 2, 5
116 115 2, 3, 5
117 180 2, 3
...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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