A131394 -id:A131394 - cp's OEIS frontend

A131396 Positions of n-th positive integer in A131394, minus 1.

1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 21, 24, 25, 27, 30, 31, 34, 35, 37, 39, 41, 43, 46, 48, 49, 51, 53, 56, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 78, 80, 82, 84, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 108, 111, 112, 114, 117, 119, 121, 123, 125, 127, 129

Offset: 1

Clark Kimberling, Jul 06 2007, Jul 21 2007

nonn

If A131394 is a permutation of the integers, as conjectured, then A131396 is the complement of A131397.

			The 5th positive integer to occur in A131393 is 6, which occurs at position 9, so that a(5) = 9 - 1 = 8.

Table of n, a(n) for n = 1..66

a(n) = position of n-th positive integers in A131393, minus 1.

A131397 Positions of negative integers in A131394, minus 1.

Original entry on oeis.org

3, 6, 7, 9, 12, 14, 15, 18, 20, 22, 23, 26, 28, 29, 32, 33, 36, 38, 40, 42, 44, 45, 47, 50, 52, 54, 55, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 79, 81, 83, 85, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 109, 110, 113, 115, 116, 118, 120, 122, 124, 126, 128, 130, 132

Offset: 1

Clark Kimberling, Jul 06 2007

nonn

If A131394 is a permutation of the integers, as conjectured, then A131397 is the complement of A131396.

			The 5th negative integer to occur in A131394 is -5, which occurs at position 13, so that a(5) = 13 - 1 = 12.

Table of n, a(n) for n = 1..65

Cf. A131388, A131389, A131390, A131391, A131392, A131393, A131394, A131395, A131396.

a(n) = position of n-th negative integer in A131394, minus 1.

A131389 Sequence (d(n)) generated by Rule 1 (see Comments) with a(1) = 1 and d(1) = 0.

Original entry on oeis.org

0, 1, 2, -1, 3, 4, -2, -3, 6, -4, 5, 7, -5, 8, -6, -7, 9, 10, -8, 11, -10, 12, -9, -11, 13, 14, -13, -12, 15, 16, -14, -15, 17, 18, -16, 19, -17, 20, -19, 21, -18, 22, -20, -21, 24, -22, 23, 25, -23, 26, -25, 27, -24, 28, -26, -27, 30, 29, -31, 32, -28, 31, -29, 33, -30, 34, -32, -33, 35, 36, -34

Offset: 1

Clark Kimberling, Jul 05 2007

sign

Rule 1 is given at A131388. Conjecture: Every integer occurs exactly once in A131389.

			See A131388.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A131388, A131390, A131391, A131392, A131393, A131394, A131395, A131396, A131397.

Mathematica
```
(* See A131388. *)
```

This is the sequence d( ) in the formula for A131388.

Definition clarified by Clark Kimberling, May 12 2015

A131388 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 0.

Original entry on oeis.org

1, 2, 4, 3, 6, 10, 8, 5, 11, 7, 12, 19, 14, 22, 16, 9, 18, 28, 20, 31, 21, 33, 24, 13, 26, 40, 27, 15, 30, 46, 32, 17, 34, 52, 36, 55, 38, 58, 39, 60, 42, 64, 44, 23, 47, 25, 48, 73, 50, 76, 51, 78, 54, 82, 56, 29, 59, 88, 57, 89, 61, 92, 63, 96, 66, 100, 68, 35, 70, 106, 72, 37

Offset: 1

Clark Kimberling, Jul 05 2007

nonn

Rule 1 follows.  For k >= 1, let  A(k) = {a(1), …, a(k)} and D(k) = {d(1), …, d(k)}.  Begin with k = 1 and nonnegative integers a(1) and d(1).
Step 1:   If there is an integer h such that 1 - a(k) < h < 0 and h is not in D(k) and a(k) + h is not in A(k), let d(k+1) be the greatest such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.
Step 2:  Let h be the least positive integer not in D(k) such that a(k) + h is not in A(k).  Let a(k+1) = a(k) + h and d(k+1) = h.  Replace k by k+1 and do Step 1.
Conjecture:  if a(1) is an nonnegative integer and d(1) is an integer, then (a(n)) is a permutation of the nonnegative integers (if a(1) = 0) or a permutation of the positive integers (if a(1) > 0).  Moreover, (d(n)) is a permutation of the integers if d(1) = 0, or of the nonzero integers if d(1) > 0.
See A257705 for a guide to related sequences.

			a(2)=1+1, a(3)=a(2)+2, a(4)=a(3)+(-1), a(5)=a(4)+3, a(6)=a(5)+4.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A131389, A131390, A131391, A131392, A131393, A131394, A131395, A131396, A131397, A257705, A175498.

(*Program 1 *)
{a, f} = {{1}, {0}}; Do[tmp = {#, # - Last[a]} &[Max[Complement[#, Intersection[a, #]] &[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
If[! IntegerQ[tmp[[1]]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f,#] && ! MemberQ[a, Last[a] + #]) &]]];
AppendTo[a, tmp[[1]]]; AppendTo[f, tmp[[2]]], {400}];
{a, f} (*{A131388, A131389}; Peter J. C. Moses, May 10 2015*)
(*Program 2 *)
a[1] = 1; d[1] = 0; k = 1; z = 10000; zz = 120;
A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
c[k_] := Complement[Range[-z, z], diff[k]];
T[k_] := -a[k] + Complement[Range[z], A[k]];
s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
Table[If[Length[s[k]] == 0, {h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}, {h = Max[s[k]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}], {i, 1, zz}];
u = Table[a[k], {k, 1, zz}] (* A131388 *)
Table[d[k], {k, 1, zz}]     (* A131389 *)

a(k+1) - a(k) = d(k+1) for k >= 1.

Revised by Clark Kimberling, May 12 2015

A131393 Conjectured permutation of the positive integers using Rule 2 with a(1)=1.

Original entry on oeis.org

1, 2, 4, 3, 6, 10, 8, 5, 11, 7, 12, 19, 14, 22, 16, 9, 18, 28, 20, 31, 21, 33, 24, 13, 26, 40, 27, 42, 30, 15, 32, 48, 34, 17, 35, 54, 38, 58, 39, 60, 37, 59, 41, 64, 44, 23, 47, 25, 50, 76, 52, 79, 53, 81, 56, 29, 61, 90, 62, 92, 63, 94, 57, 91, 55, 88, 49, 84, 51, 87, 46, 83

Offset: 1

Clark Kimberling, Jul 05 2007

Conjecture 1: a( ) is a permutation of the positive integers. Conjecture 2: d( ) is a permutation of the integers. The sequence using Rule 1 ("negative before positive") is A131388.

This sequence was generated using "Rule 2" in a computer program which been lost. The wording of "Rule 2" in the Formula section, although flawed, is retained in case someone can rediscover "Rule 2" and contribute a corrected version. - Clark Kimberling, May 18 2015

			a(2)=1+1, a(3)=a(2)+2, a(4)=a(3)+(-1), a(5)=a(4)+3, a(6)=a(5)+4.
The first term that differs from A131388 is a(28)=42.

Table of n, a(n) for n = 1..72

Cf. A131388, A131389, A131390, A131391, A131392, A131394, A131395, A131396, A131397.

The following version of "Rule 2" is defective; see Comments. - Clark Kimberling, May 18 2015

Rule 2 ("positive before negative"): define sequences d( ) and a( ) as follows: d(1)=0, a(1)=1 and for n>=2, d(n) is the least positive integer d such that a(n-1)+d is not among a(1), a(2),...,a(n-1), or, if no such d exists, then d(n) is the greatest negative integer d such that a(n-1)+d is not among a(1), a(2),...,a(n-1). Then a(n)=a(n-1)+d.

A131391 Positions of positive integers in A131389, minus 1.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 21, 24, 25, 28, 29, 32, 33, 35, 37, 39, 41, 44, 46, 47, 49, 51, 53, 56, 57, 59, 61, 63, 65, 68, 69, 72, 73, 75, 77, 79, 81, 83, 85, 88, 90, 92, 94, 95, 97, 99, 101, 103, 105, 108, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129

Offset: 1

Clark Kimberling, Jul 06 2007

nonn

If A131389 is a permutation of the integers, as conjectured, then A131391 is the complement of A131392.

			The 5th positive integer to occur in A131391 is 6, which occurs in position 9, so that a(5) = 9 - 1 = 8.

Cf. A131388, A131389, A131390, A131392, A131393, A131394, A131395, A131396, A131397.

a(n) = -1 + position of the n-th positive integer in the sequence A131389=(0,1,2,-1,3,4,-2,-3,6,-4,5,...)

A131392 Positions of negative integers in A131389, minus 1.

Original entry on oeis.org

3, 6, 7, 9, 12, 14, 15, 18, 20, 22, 23, 26, 27, 30, 31, 34, 36, 38, 40, 42, 43, 45, 48, 50, 52, 54, 55, 58, 60, 62, 64, 66, 67, 70, 71, 74, 76, 78, 80, 82, 84, 86, 87, 89, 91, 93, 96, 98, 100, 102, 104, 106, 107, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 131, 132

Offset: 1

Clark Kimberling, Jul 06 2007

nonn

If A131389 is a permutation of the integers, as conjectured, then A131392 is the complement of A131391.

			The 5th negative integer to occur in A131391 is -5, which occurs in position 13, so that a(5)=13-1=12.

Cf. A131388, A131389, A131390, A131391, A131393, A131394, A131395, A131396, A131397.

a(n) = -1 + position of the n-th negative integer in the sequence A131389=(0,1,2,-1,3,4,-2,-3,6,-4,5,...)

A131390 Conjectured permutation of the positive integers; inverse of conjectured permutation A131388.

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 10, 7, 16, 6, 9, 11, 24, 13, 28, 15, 32, 17, 12, 19, 21, 14, 44, 23, 46, 25, 27, 18, 56, 29, 20, 31, 22, 33, 68, 35, 72, 37, 39, 26, 90, 41, 88, 43, 92, 30, 45, 47, 94, 49, 51, 34, 108, 53, 36, 55, 59, 38, 57, 40, 61, 63, 42, 81, 65, 83, 67, 79, 69, 77, 71, 48

Offset: 1

Clark Kimberling, Jul 05 2007

nonn

Cf. A131388, A131389, A131391, A131392, A131393, A131394, A131395, A131396, A131397.

Inverse of A131388.

A131395 Conjectured permutation of the positive integers; inverse of conjectured permutation A131393.

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 10, 7, 16, 6, 9, 11, 24, 13, 30, 15, 34, 17, 12, 19, 21, 14, 46, 23, 48, 25, 27, 18, 56, 29, 20, 31, 22, 33, 35, 77, 41, 37, 39, 26, 43, 28, 75, 45, 73, 71, 47, 32, 67, 49, 69, 51, 53, 36, 65, 55, 63, 38, 42, 40, 57, 59, 61, 44, 99, 97, 87, 93, 78, 91, 50, 89

Offset: 1

Clark Kimberling, Jul 05 2007

nonn

Cf. A131388, A131389, A131390, A131391, A131392, A131393, A131394, A131396, A131397.

Inverse of A131393.

A257985 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 2 and d(1) = 0.

Original entry on oeis.org

2, 3, 5, 4, 7, 13, 6, 11, 8, 15, 9, 17, 12, 10, 19, 37, 14, 27, 16, 31, 18, 35, 20, 39, 21, 41, 22, 43, 23, 45, 24, 47, 25, 49, 32, 28, 55, 26, 51, 42, 30, 59, 29, 57, 33, 65, 34, 67, 40, 44, 36, 71, 38, 75, 50, 99, 46, 91, 48, 95, 53, 105, 54, 107, 52, 103

Offset: 1

Clark Kimberling, Jun 02 2015

Rule 3 follows.  For k >= 1, let  A(k) = {a(1), …, a(k)} and D(k) = {d(1), …, d(k)}.  Begin with k = 1 and nonnegative integers a(1) and d(1).
Step 1:   If there is an integer h such that 1 - a(k) < h < 0 and h is not in D(k) and a(k) + h is not in A(k), let d(k+1) be the least such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.
Step 2:  Let h be the least positive integer not in D(k) such that a(k) - h is not in A(k).  Let a(k+1) = a(k) + h and d(k+1) = h.  Replace k by k+1 and do Step 1.
See A257905 for a guide to related sequences and conjectures.

			a(1) = 2, d(1) = 0;
a(2) = 3, d(2) = 1;
a(3) = 5, d(3) = 2;
a(4) = 4, d(4) = -1.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A257905, A257986, A131394.

{a, f} = {{2}, {0}}; Do[tmp = {#, # - Last[a]} &[Min[Complement[#, Intersection[a, #]]&[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
If[! IntegerQ[tmp[[1]]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f, #] && ! MemberQ[a, Last[a] - #]) &]]]; AppendTo[a, tmp[[1]]]; AppendTo[f, tmp[[2]]], {120}]; {a, f} (* Peter J. C. Moses, May 14 2015 *)

a(n) = A131393(n) + 1 for n >= 1. Also, a(n) - a(n-1) = A131394(n) for n >= 2.

cp's OEIS Frontend

A131396 Positions of n-th positive integer in A131394, minus 1.

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A131397 Positions of negative integers in A131394, minus 1.

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A131389 Sequence (d(n)) generated by Rule 1 (see Comments) with a(1) = 1 and d(1) = 0.

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A131388 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 0.

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A131393 Conjectured permutation of the positive integers using Rule 2 with a(1)=1.

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A131391 Positions of positive integers in A131389, minus 1.

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A131392 Positions of negative integers in A131389, minus 1.

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A131390 Conjectured permutation of the positive integers; inverse of conjectured permutation A131388.

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A131395 Conjectured permutation of the positive integers; inverse of conjectured permutation A131393.

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A257985 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 2 and d(1) = 0.