A367832 -id:A367832 - cp's OEIS frontend

A367467 Lexicographically earliest infinite sequence of positive integers such that a(n + a(n)) is distinct for all n.

1, 1, 2, 2, 3, 4, 2, 5, 6, 7, 1, 8, 9, 2, 10, 11, 12, 1, 13, 14, 2, 15, 16, 2, 17, 18, 19, 2, 20, 21, 2, 22, 23, 24, 1, 25, 26, 2, 27, 28, 2, 29, 30, 31, 2, 32, 33, 2, 34, 35, 36, 1, 37, 38, 2, 39, 40, 41, 1, 42, 43, 2, 44, 45, 2, 46, 47, 48, 1, 49, 50, 2, 51, 52, 53, 1, 54, 55, 2, 56, 57, 2, 58, 59, 60, 2, 61, 62, 2

Offset: 1

Neal Gersh Tolunsky, Nov 18 2023

nonn

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 previous location.
2. Location n reaches a location in one jump that is not reached in one jump from a location before n.
Described in the above way, the sequence seems to be structured as follows:
A083051 appears to give the indices which cannot be reached from any earlier term; the terms at these indices are 1s and 2s.
A087057 appears to give the indices which can be reached from an earlier term; except for a(2), these terms are first occurrences.
From Thomas Scheuerle, Nov 26 2023: (Start)
Empirical observations:
It appears that this sequence consists of the natural numbers in ascending order interspersed by 1 and 2.
If we consider the distance between successive ones, we will observe a nonperiodic pattern: 9,7,17,17,7,10,7,17,7,10,... . It appears that there are only 7, 10 and 17 with the exception of 9 once.
If we consider the distance between successive twos, we will also observe an interesting nonperiodic pattern: 3,7,7,3,4,3,7,3,4,3,7,7,3,... . It appears that this pattern consists only of 3, 4 and 7. (End)

			Initial locations and the (by definition) distinct terms that they reach:
     n|  1  2  3  4  5  6  7  8  9
  a(n)|  1  1  2  2  3  4  2  5  6
          =>1=>2====>3
                   ====>4
                      =======>5
                            ====>6
When we evaluate a(i+a(i)) with each index i, we get a distinct value. When i=1, for example, a(1+a(1))=a(1+1)=a(2)=1;  no other i gives 1 as the solution to a(i+a(i)). When i=4, a(4+a(4))=a(4+2)=a(6)=4, and 4 is likewise a solution unique to i=4.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..1000

Cf. A367832, A293078, A323420, A359807, A367039, A337226, A366691.

Cf. A049472, A083051, A087057, A184119, A328987.

function a = A367467( max_n )
    a = [1 1:2*max_n];
    for n = 3:max_n
        a(n) = 1;
        while consistency(a, n) == false
            a(n) = a(n)+1;
        end
    end
    a = a(1:max_n);
end
function ok = consistency(a, n)
    v = a([1:n] + a(1:n));
    ok = (n == length(unique(v)));
end % Thomas Scheuerle, Nov 21 2023

From Thomas Scheuerle, Nov 26 2023: (Start)
Conjectures:
a(n) = A049472(n) = floor(n*(1 + 1/sqrt(2))) - n, if n is not in A083051.
a(A083051(n)) = A184119(n+1) - A083051(n).
a(a(A083051(n)) + A083051(n)) + a(A083051(n)) + A083051(n) = A328987(n) = floor((a(A083051(n)) + A083051(n))*(1 + 1/sqrt(2))) = floor(A184119(n+1)*(1 + 1/sqrt(2))). (End)

A367849 Lexicographically earliest infinite sequence of positive integers such that for each n, the values in a path of locations starting from any i=n are all distinct, where jumps are allowed from location i to i+a(i).

Original entry on oeis.org

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

Offset: 1

Neal Gersh Tolunsky, Dec 08 2023

nonn

Consider each index i as a location from which one can jump a(i) terms forward. No starting index can reach the same value more than once by forward jumps.
The value a(i) at the starting index is not part of the path (and thus allows a(2)=1).
The indices of first occurrences are given by A133263 (essentially triangular numbers + 2).
Changing the definition so that jumps are allowed only from location i to i-a(i) gives A002260.

			We can see, for example, that the terms reachable by jumping forward continuously from i=1 are all distinct (and in this case are just the positive integers):
  1, 1, 2, 1, 3, 1, 1, 4, 1, 1, 2, 5
  *->1->2---->3------->4---------->5
Beginning at i=9 and jumping forward continuously, we get the sequence 1,2,3,4,5,6,7,9 (in which all terms are likewise distinct).

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, Plot of a(1..100000)^2. It is conjectured that only the topmost straight line in this plot will extend into infinity.
Thomas Scheuerle, Partial view of a(1..80000)^2 equalized to Y axis by shear mapping. This shows the different lengths of the increasing sequences.
Neal Gersh Tolunsky, Ordinal transform of 20000 terms
Neal Gersh Tolunsky, First differences of 20000 terms

Cf. A367467, A367832, A133263 (index of first occurrences), A362248.

function a = A367849( max_n )
    a = zeros(1,max_n); j = find(a == 0,1);
    while ~isempty(j)
        a(j) = 1; k = 1;
        if j+k < max_n
            while a(j+k) == 0
                a(j+k) = k;
                if j+2*k-1 < max_n
                    j = j+(k-1); k = k+1;
                else
                    break;
                end
            end
        end
        j = find(a == 0,1);
    end
end % Thomas Scheuerle, Dec 09 2023

a(A133263(n)) = n + 1.

A368485 Lexicographically earliest infinite sequence of positive integers such that for n > 1, a(n - a(n)) is distinct for all indices n with the same a(n) value.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 4, 4, 5, 6, 1, 2, 3, 4, 5, 5, 5, 6, 7, 1, 2, 3, 4, 5, 6, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 6, 6, 6, 7, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 7, 7, 7, 8, 8, 9, 10, 11

Offset: 1

Neal Gersh Tolunsky, Dec 28 2023

nonn

Consider each index i as a location from which one can jump a(i) terms backwards. Any two distinct indices m and k, where a(m)=a(k), will jump to distinct values. In other words, every 1 will jump back to a distinct value, every 2, 3, etc.

			We can see, for example, that the values reached by jumping backwards once from each 3 in the sequence are all distinct:
  1, 1, 2, 1, 2, 3, 1, 2, 3, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3
        2<=======*
                 3<=======*
                    1<=======*  4<=======*     5<=======*

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

Cf. A367849, A367832, A367467.

PARI
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See Links section.
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More terms from Rémy Sigrist, Jan 15 2024

A369475 Lexicographically earliest infinite sequence such that, from all indices n with the same a(n) value, the terms reached by a single jump are all distinct, where jumps are allowed from location i to i+-a(i).

Original entry on oeis.org

1, 2, 2, 3, 4, 1, 5, 3, 2, 5, 6, 1, 7, 4, 6, 3, 1, 8, 8, 2, 5, 7, 3, 5, 6, 9, 1, 10, 11, 1, 12, 3, 2, 3, 10, 4, 13, 1, 14, 6, 2, 3, 9, 5, 15, 7, 2, 9, 13, 7, 5, 4, 4, 4, 6, 10, 12, 11, 9, 2, 10, 16, 1, 15, 3, 4, 5, 17, 1, 18, 9, 12, 3, 6, 5, 19, 1, 20, 9, 15

Offset: 1

Neal Gersh Tolunsky, Jan 23 2024

Consider each index i as a location from which one can jump a(i) terms forwards or backwards. From all indices with the same a(n) value, every jump is to a distinct term.

Another way to define the sequence is to consider every possible ordered pair of values of the form (origin value, destination value)--every such ordered pair is distinct.

			a(5)=4 because:
a(5) cannot be 1 because then we would have two jumps from a term with the same value 2, both landing on the value 1--ordered pair (2,1) twice:
  1, 2, 2, 3, 1
        2---->1
  1<----2
a(5) cannot be 2 because we would have two jumps from the same a(n) value 2 to the same value 2--ordered pair (2,2) twice:
  1, 2, 2, 3, 2
        2---->2
        2<----2
a(5) cannot be 3 because we would have two jumps from the same a(n) value 2 to the same a(n) value 3--ordered pair (2,3) twice:
  1, 2, 2, 3, 3
     2---->3
        2---->3
a(5) can be 4 without contradiction.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Pontus von Brömssen, Plot of (n,a(n)) for n = 1..1000000.
Pontus von Brömssen, Plot of (n, log(a(n))/log(n)) for n = 2..1000000.

Cf. A368485, A367849, A367832, A367467.

More terms from Pontus von Brömssen, Jan 24 2024

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).

Original entry on oeis.org

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

Neal Gersh Tolunsky, Feb 06 2024

nonn

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.

			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

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

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

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 *)

More terms from Giorgos Kalogeropoulos, Feb 28 2024

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A367467 Lexicographically earliest infinite sequence of positive integers such that a(n + a(n)) is distinct for all n.

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A367849 Lexicographically earliest infinite sequence of positive integers such that for each n, the values in a path of locations starting from any i=n are all distinct, where jumps are allowed from location i to i+a(i).

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A368485 Lexicographically earliest infinite sequence of positive integers such that for n > 1, a(n - a(n)) is distinct for all indices n with the same a(n) value.

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A369475 Lexicographically earliest infinite sequence such that, from all indices n with the same a(n) value, the terms reached by a single jump are all distinct, where jumps are allowed from location i to i+-a(i).

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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).

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