A375834 -id:A375834 - cp's OEIS frontend

A375814 Lexicographically earliest sequence of positive integers such that for any n > 0, Sum_{k = 1..n} 1/(a(k)*a(n+1-k)) < 1.

2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 6, 7, 8, 7, 8, 8, 7, 9, 8, 9, 9, 8, 10, 9, 9, 10, 10, 10, 10, 10, 10, 11, 10, 11, 11, 11, 12, 11, 11, 13, 11, 12, 12, 12, 13, 12, 12, 14, 12, 13, 14, 12, 14, 14, 13, 14, 13, 14, 15, 13, 15, 15, 13, 16, 14, 14, 16

Offset: 1

Rémy Sigrist, Aug 30 2024

Sum_{k = 2..300} 1/(a(k)*a(302-k)) > 1, hence the sequence is finite with 300 terms.

			The first terms, alongside the corresponding sums, are:
  n   a(n)  Sum_{k=1..n} 1/(a(k)*a(n+1-k))
  --  ----  ------------------------------
   1     2  1/4
   2     2  1/2
   3     2  3/4
   4     3  5/6
   5     3  11/12
   6     4  11/12
   7     4  17/18
   8     4  35/36
   9     5  44/45
  10     5  59/60
  11     5  239/240
  12     6  119/120
  13     6  79/80
  14     6  119/120
  15     7  991/1008

Rémy Sigrist, Table of n, a(n) for n = 1..300

Cf. A375815, A375834.

{ for (n = 1, #a = vector(72), if (n==1, a[n] = 2, x = sum(k = 2, n-1, 1/(a[k]*a[n+1-k])); if (x >= 1, break, a[n] = floor(2/(a[1]*(1-x)))+1;);); print1 (a[n]", ");); }

A375815 Lexicographically earliest sequence of positive integers such that for any n > 0, Sum_{k = 1..n} 1/(a(k)*a(n+1-k)) <= 1.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 14, 15, 15, 15

Offset: 1

Rémy Sigrist, Aug 30 2024

nonn

The variant with strict inequality (A375814) is finite; is this sequence infinite?

			The first terms, alongside the corresponding sums, are:
  n   a(n)  Sum {k=1..n} 1/(a(k)*a(n+1-k))
  --  ----  ------------------------------
   1     1  1
   2     2  1
   3     3  11/12
   4     3  1
   5     4  17/18
   6     4  35/36
   7     5  167/180
   8     5  14/15
   9     5  77/80
  10     5  119/120
  11     6  77/80
  12     6  29/30
  13     6  443/450
  14     7  3007/3150
  15     7  6011/6300

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A375814, A375834.

{ for (n = 1, #a = vector(72), if (n==1, a[n] = 1, x = sum(k = 2, n-1, 1/(a[k]*a[n+1-k])); if (x >= 1, break, a[n] = ceil(2/(a[1]*(1-x))););); print1 (a[n]", ");); }

cp's OEIS Frontend

A375814 Lexicographically earliest sequence of positive integers such that for any n > 0, Sum_{k = 1..n} 1/(a(k)*a(n+1-k)) < 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI