A030194 -id:A030194 - cp's OEIS frontend

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, 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

Offset: 1

Lechoslaw Ratajczak, Oct 04 2020

nonn

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?

			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.

Cf. A000217, A030194.

a(n) = A030194(n-1) + 1. - Hugo Pfoertner, Oct 05 2020

A359246 Lexicographically earliest sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.

Original entry on oeis.org

2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2, 20, 2, 7, 2, 7, 2, 14, 9, 7, 2, 7, 2, 29, 2, 7, 2, 7, 2, 7, 2, 41, 9, 9, 16, 22, 2, 23, 7, 2, 7, 2, 7, 2, 7, 22, 9, 2, 7, 2, 7, 43, 9, 29, 2, 41, 9, 7, 2, 9, 5, 2, 7, 2, 22, 9, 9, 9, 25, 9, 29, 2, 7, 2, 7, 2, 32, 43, 65, 5, 2, 2

Offset: 0

Ctibor O. Zizka, Jan 21 2023

nonn

Differences of A030194.

			a(0) = 2 by the definition of the sequence. The next number >= 2 is 2; {2, 2 + 2} are not triangular numbers, thus a(1) = 2. Then we try 2; but 2 + 2 + 2 is a triangular number. We cannot try 3, which is a triangular number, so we try 4; but 4 + 2 is a triangular number, so we try 5; {5, 5 + 2, 5 + 2 + 2} are not triangular numbers, thus a(2) = 5.

Cf. A000217, A030194 (partial sums), A332941, A360019.

q:= proc(n) option remember; issqr(8*n+1) end:
s:= proc(i, j) option remember; `if`(i>j, 0, a(j)+s(i, j-1)) end:
a:= proc(n) option remember; local k; for k from 2 while
      ormap(q, [k+s(i, n-1)$i=0..n]) do od; k
    end: a(-1):=-1:
seq(a(n), n=0..81);  # Alois P. Heinz, Jan 21 2023

triQ[n_] := IntegerQ @ Sqrt[8*n + 1]; a[0] = 2; a[n_] := a[n] = Module[{k = 2, t = Accumulate @ Table[a[i], {i, n - 1, 0, -1}]}, While[triQ[k] || AnyTrue[t + k, triQ], k++]; k]; Array[a, 100, 0] (* Amiram Eldar, Jan 21 2023 *)

More terms from Amiram Eldar, Jan 21 2023

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A359246 Lexicographically earliest sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.

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