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A280659
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Lexicographically earliest sequence of distinct positive terms such that the sum of two consecutive terms has at least 5 distinct prime factors.
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2
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1, 2309, 421, 1889, 841, 1469, 1261, 1049, 1681, 629, 2101, 209, 2521, 1769, 541, 2189, 121, 2609, 961, 1349, 1381, 929, 1801, 509, 2221, 89, 2641, 1649, 661, 2069, 241, 2489, 1081, 1229, 1501, 809, 1921, 389, 2341, 1949, 361, 2369, 1201, 1109, 1621, 689, 2041
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OFFSET
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1,2
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COMMENTS
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Conjecturally: this sequence is a permutation of the natural numbers, and a(n) ~ n.
The first fixed points are: 1, 7379, 7730, 7765, 7846, 9535, 9903, 11604, 11631, 11741, 12674, 15549, 15824, 16670, 16745, 16800, 16806, 16841.
This sequence has similarities with A285487: here we consider the sum of consecutive terms, there the product of consecutive terms.
The scatterplot of the first terms presents rectangular clusters of points near the origin; these clusters seem to correspond to indexes n satisfying a(n) + a(n+1) < 2 * prime#(5) (where prime(k)# = A002110(k)).
Near the origin, we also have ranges of more than hundred consecutive terms where the function b satisfying b(n) = lpf(a(n)) (where lpf = A020639) is constant (and equals 2, 3 or 5).
These features are highlighted in the alternate scatterplots provided in the Links section.
There features are also visible in the scatterplots of variants of this sequence where we increase the minimum number of distinct prime factors required for the sum of two consecutive terms.
(End)
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LINKS
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EXAMPLE
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The first terms, alongside the primes p dividing a(n)+a(n+1), are:
n a(n) p
-- ---- --------------
1 1 2, 3, 5, 7, 11
2 2309 2, 3, 5, 7, 13
3 421 2, 3, 5, 7, 11
4 1889 2, 3, 5, 7, 13
5 841 2, 3, 5, 7, 11
6 1469 2, 3, 5, 7, 13
7 1261 2, 3, 5, 7, 11
8 1049 2, 3, 5, 7, 13
9 1681 2, 3, 5, 7, 11
10 629 2, 3, 5, 7, 13
11 2101 2, 3, 5, 7, 11
12 209 2, 3, 5, 7, 13
13 2521 2, 3, 5, 11, 13
14 1769 2, 3, 5, 7, 11
15 541 2, 3, 5, 7, 13
16 2189 2, 3, 5, 7, 11
17 121 2, 3, 5, 7, 13
18 2609 2, 3, 5, 7, 17
19 961 2, 3, 5, 7, 11
20 1349 2, 3, 5, 7, 13
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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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