A330241 a(n) is the greatest k such that there is an increasing sequence of positive integers j(0),j(1),...,j(k) such that n == i (mod j(i)) for each i.

0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 3, 4, 1, 2, 1, 2, 3, 4, 3, 4, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 3, 4, 3, 4, 1, 2, 3, 4, 5, 6, 3, 4, 1

Offset

0,6

Comments

If n == -1 (mod A003418(k)), then a(n) == j mod (j+1) for j <= k-1, so a(n) >= k-1. In particular, the sequence is unbounded.

a(n) = 1 if n is in A008864.

a(n+1) <= a(n)+1, with equality if n is even.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Example

a(5) = 2 because 5 == 0 (mod 1), == 1 (mod 2), == 2 (mod 3), but there is no j > 3 with 5 == 3 (mod j).
a(11) = 4 because 11 == 0 (mod 1), == 1 (mod 2), == 2 (mod 3), == 3 (mod 4), == 4 (mod 7), but there is no j > 7 with 11 == 5 (mod j).

Maple

F:= proc(m) local L, j, k;
  L:= [seq(m mod j, j=1..m-1)];
  k:= 0; j:= 1;
  for k from 1 do
    do j:= j+1;
      if j = m then return k-1 fi;
    until L[j] = k
  od;
end proc:
F(0):= 0:
F(1):= 1:
map(F, [$0..100]);

Crossrefs

Cf. A003418, A008864, A329773.

Sequence in context: A161271 A160975 A305300 * A175851 A049711 A137293

Adjacent sequences: A330238 A330239 A330240 * A330242 A330243 A330244

Keyword

nonn,hear

Author

J. M. Bergot and Robert Israel, Dec 06 2019

Status

approved