A369852 a(1)=1, a(2)=2; thereafter, any two indices n with different a(n) values reach distinct values by a single jump, where jumps are allowed from location i to i+a(i).

1, 2, 2, 3, 1, 2, 4, 1, 5, 2, 6, 1, 2, 7, 1, 2, 8, 1, 5, 2, 9, 1, 5, 7, 10, 1, 2, 11, 3, 12, 9, 4, 1, 13, 14, 15, 1, 5, 16, 12, 3, 17, 7, 4, 1, 18, 19, 6, 20, 21, 22, 23, 3, 8, 24, 4, 1, 5, 25, 26, 4, 10, 7, 27, 15, 28, 1, 13, 29, 30, 31, 32, 33, 2, 34, 1, 5, 5

Offset: 1

Author

Neal Gersh Tolunsky, Feb 06 2024

Keywords

• nonn

Comments

Consider each index i as a location from which one can jump a(i) terms forward. To find a(n) we have to check 2 conditions:

1. The value a(n) can be reached in one jump by at most one distinct value.

2. Location n reaches a location in one jump that is not reached in one jump from a location before n.

Another way to view the sequence is to consider the sets of values that can be reached from each distinct integer by a single jump forward (values reached by 1s in the sequence, values reached by 2s, 3s etc.): all of these sets are disjoint.

Examples

			a(4)=3 because:
  a(4) cannot be 1 because then we would have two distinct values (a(3)=2, a(4)=1) that reach the same future value a(5)=x:
  1, 2, 2, 1, x
        2---->x
           1->x
  a(4) cannot be 2 because then we would have two distinct values (a(1)=1, a(2)=2) reach the same value 2:
  1, 2, 2, 2
  1->2
     2---->2
  a(4) can be 3 without contradiction since there is only one distinct value that can reach the value 3 (a(2)=2):
  1, 2, 2, 3
     2---->3
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A369475, A368485, A367849, A367832, A367467.

Programs

Mathematica

lst={1,2};Do[z=1;Quiet@While[l=Join[lst,{z}]; Union[Length@*Union/@ GatherBy[Select[Table[{l[[k]],l[[l[[k]]+k]]},{k,Length@l}],IntegerQ@Last@#&],Last]]!={1}||
MemberQ[Table[l[[k]]+k,{k,Length@l-1}],Length@l+Last@l],z++];AppendTo[lst,z],{i,89}];lst (* Giorgos Kalogeropoulos, Feb 29 2024 *)

Extensions

More terms from Giorgos Kalogeropoulos, Feb 28 2024