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A297076
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Lexicographically earliest sequence of distinct positive numbers such that, for any n > 1, the prime factorizations of a(n) and of a(n+1) share a prime power > 1.
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1
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1, 2, 6, 3, 12, 4, 20, 5, 10, 14, 7, 21, 15, 24, 8, 40, 30, 18, 9, 36, 28, 35, 42, 22, 11, 33, 39, 13, 26, 34, 17, 51, 48, 16, 80, 45, 55, 44, 52, 60, 57, 19, 38, 46, 23, 69, 66, 50, 25, 75, 78, 54, 27, 108, 68, 76, 84, 56, 63, 70, 58, 29, 87, 93, 31, 62, 74
(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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For any n > 1, there is a prime number p that divides a(n) and such that the p-adic valuation of a(n) equals the p-adic valuation of a(n+1).
This sequence has similarities with the EKG sequence A064413: here consecutive terms share a prime power, there consecutive terms share a prime factor.
The scatterplots of this sequence and of A064413 show three similar lines starting from the origin; however here we also have some isolated points.
This sequence and A064413 match for n = 1, 2, 9, 28, 78, 113, 245, 419, 420, 421, 580, 581, 582, 644, 816, 937, 969, 1105, 1117, 1118, etc.
This sequence is likely to be a permutation of the natural numbers.
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LINKS
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EXAMPLE
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The first terms, alongside their prime factorizations, are:
n a(n) Prime factorization of a(n)
-- ---- ---------------------------
1 1 1
2 2 2
3 6 2 * 3
4 3 3
5 12 2^2 * 3
6 4 2^2
7 20 2^2 * 5
8 5 5
9 10 2 * 5
10 14 2 * 7
11 7 7
12 21 3 * 7
13 15 3 * 5
14 24 2^3 * 3
15 8 2^3
16 40 2^3 * 5
17 30 2 * 3 * 5
18 18 2 * 3^2
19 9 3^2
20 36 2^2 * 3^2
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PROG
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(PARI) See Links section.
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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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