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A336531 A sieve: start with the positive integers. Let a(1)=1. Mark out the following numbers: a(1)+1, a(1)+1+2, a(1)+1+2+3, a(1)+1+2+3+4, ... . The next integer in the list not marked out is 3, so a(2)=3. Mark out the following numbers: a(2)+1, a(2)+1+2, a(2)+1+2+3, a(2)+1+2+3+4, ... . Repeat the procedure for a(3), a(4), a(5), ... . 1
1, 3, 5, 10, 12, 14, 19, 21, 23, 28, 30, 32, 52, 54, 61, 63, 70, 72, 86, 95, 102, 104, 111, 113, 142, 144, 151, 153, 160, 162, 169, 171, 212, 221, 230, 246, 268, 270, 293, 300, 302, 309, 311, 318, 320, 327, 349, 358, 360 (list; graph; refs; listen; history; text; internal format)
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:

---------------------

  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

---------------------

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?

LINKS

Table of n, a(n) for n=1..49.

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

Cf. A000217, A030194.

Sequence in context: A075741 A119133 A284959 * A285415 A275668 A212537

Adjacent sequences:  A336528 A336529 A336530 * A336532 A336533 A336534

KEYWORD

nonn

AUTHOR

Lechoslaw Ratajczak, Oct 04 2020

STATUS

approved

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Last modified January 19 00:22 EST 2022. Contains 350464 sequences. (Running on oeis4.)