A280659 Lexicographically earliest sequence of distinct positive terms such that the sum of two consecutive terms has at least 5 distinct prime factors.

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

Offset: 1

Author

Rémy Sigrist, Apr 25 2017

Keywords

• nonn
• look

Comments

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.

From Rémy Sigrist, Jul 16 2017: (Start)

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)

Examples

			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
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A280659
• Rémy Sigrist, Scatterplot of the first 10000 terms, highlighting the rectangular clusters near the origin
• Rémy Sigrist, Scatterplot of the first 10000 terms, highlighting lpf(a(n)) = 2, 3 or 5

Crossrefs

Cf. A002110, A020639, A285487.