OFFSET
1,2
COMMENTS
Are there infinitely many pairs of the form (a(n), a(n)+2)? Let b(m) be the number of pairs less than m that differ by 2, and let s be the sum of reciprocals of consecutive terms of these pairs:
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m | b(m)| s
---------------------
10^2 | 11 | 2.627931
10^3 | 34 | 2.788503
10^4 | 64 | 2.807758
10^5 | 95 | 2.809793
10^6 | 151 | 2.810210
10^7 | 241 | 2.810273
10^8 | 386 | 2.810284
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Does the sum of these reciprocals ((1/1 + 1/3) +(1/3 + 1/5) + (1/10 + 1/12) + (1/12 + 1/14) + (1/19 + 1/21) + ...) converge to a finite number?
FORMULA
a(n) = A030194(n-1) + 1. - Hugo Pfoertner, Oct 05 2020
EXAMPLE
The first few sieving stages are as follows:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ...
1 X 3 X 5 6 X 8 9 10 X 12 13 14 15 X 17 18 19 20 21 X 23 ...
1 X 3 XX 5 X X 8 X 10 X 12 X 14 15 X 17 X 19 20 21 X 23 ...
1 X 3 XX 5 XX X X X 10 XX 12 X 14 X X 17 X 19 X 21 X 23 ...
1 X 3 XX 5 XX X X X 10 XXX 12 XX 14 X XX 17 X 19 XX 21 X 23 ...
1 X 3 XX 5 XX X X X 10 XXX 12 XXX 14 XX XX 17 XX 19 XX 21 XX 23 ...
1 X 3 XX 5 XX X X X 10 XXX 12 XXX 14 XXX XX X XX 19 XXX 21 XX 23 ...
1 X 3 XX 5 XX X X X 10 XXX 12 XXX 14 XXX XX X XX 19 XXXX 21 XXX 23 ...
1 X 3 XX 5 XX X X X 10 XXX 12 XXX 14 XXX XX X XX 19 XXXX 21 XXXX 23 ...
... Continue forever and the numbers not crossed off give the sequence.
CROSSREFS
KEYWORD
nonn
AUTHOR
Lechoslaw Ratajczak, Oct 04 2020
STATUS
approved