This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336531 #14 Oct 05 2020 05:56:37 %S A336531 1,3,5,10,12,14,19,21,23,28,30,32,52,54,61,63,70,72,86,95,102,104,111, %T A336531 113,142,144,151,153,160,162,169,171,212,221,230,246,268,270,293,300, %U A336531 302,309,311,318,320,327,349,358,360 %N 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), ... . %C A336531 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: %C A336531 --------------------- %C A336531 m | b(m)| s %C A336531 --------------------- %C A336531 10^2 | 11 | 2.627931 %C A336531 10^3 | 34 | 2.788503 %C A336531 10^4 | 64 | 2.807758 %C A336531 10^5 | 95 | 2.809793 %C A336531 10^6 | 151 | 2.810210 %C A336531 10^7 | 241 | 2.810273 %C A336531 10^8 | 386 | 2.810284 %C A336531 --------------------- %C A336531 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? %F A336531 a(n) = A030194(n-1) + 1. - _Hugo Pfoertner_, Oct 05 2020 %e A336531 The first few sieving stages are as follows: %e A336531 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ... %e A336531 1 X 3 X 5 6 X 8 9 10 X 12 13 14 15 X 17 18 19 20 21 X 23 ... %e A336531 1 X 3 XX 5 X X 8 X 10 X 12 X 14 15 X 17 X 19 20 21 X 23 ... %e A336531 1 X 3 XX 5 XX X X X 10 XX 12 X 14 X X 17 X 19 X 21 X 23 ... %e A336531 1 X 3 XX 5 XX X X X 10 XXX 12 XX 14 X XX 17 X 19 XX 21 X 23 ... %e A336531 1 X 3 XX 5 XX X X X 10 XXX 12 XXX 14 XX XX 17 XX 19 XX 21 XX 23 ... %e A336531 1 X 3 XX 5 XX X X X 10 XXX 12 XXX 14 XXX XX X XX 19 XXX 21 XX 23 ... %e A336531 1 X 3 XX 5 XX X X X 10 XXX 12 XXX 14 XXX XX X XX 19 XXXX 21 XXX 23 ... %e A336531 1 X 3 XX 5 XX X X X 10 XXX 12 XXX 14 XXX XX X XX 19 XXXX 21 XXXX 23 ... %e A336531 ... Continue forever and the numbers not crossed off give the sequence. %Y A336531 Cf. A000217, A030194. %K A336531 nonn %O A336531 1,2 %A A336531 _Lechoslaw Ratajczak_, Oct 04 2020