A175499 -id:A175499 - cp's OEIS frontend

A175498 a(1)=1. a(n) = the smallest positive integer not occurring earlier such that a(n)-a(n-1) doesn't equal a(k)-a(k-1) for any k with 2 <= k <= n-1.

1, 2, 4, 3, 6, 10, 5, 11, 7, 12, 9, 16, 8, 17, 15, 23, 13, 24, 18, 28, 14, 26, 19, 32, 20, 34, 21, 36, 25, 41, 22, 39, 30, 48, 27, 46, 29, 49, 31, 52, 37, 59, 33, 56, 40, 64, 35, 60, 38, 65, 42, 68, 43, 71, 44, 73, 45, 75, 51, 82, 47, 79, 112, 50, 84, 53, 88, 54, 90, 57, 94, 55, 93, 61, 100, 58, 98, 62, 103, 63, 105, 67

Offset: 1

Leroy Quet, May 31 2010

This sequence is a permutation of the positive integers.
a(n+1)-a(n) = A175499(n).
Conjecture: the lexicographically earliest permutation of {1,2,...n} for which differences of adjacent numbers are all distinct (cf. A131529) has, for n-->infinity, this sequence as its prefix. - Joerg Arndt, May 27 2012

Joerg Arndt and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000, first 1122 terms from Joerg Arndt
Index entries for sequences that are permutations of the natural numbers

Cf. A081145, A175499, A257465 (inverse), A257883, A131388, A131389, A257705.

import Data.List (delete)
a175498 n = a175498_list !! (n-1)
a175498_list = 1 : f 1 [2..] [] where
   f x zs ds = g zs where
     g (y:ys) | diff `elem` ds = g ys
              | otherwise      = y : f y (delete y zs) (diff:ds)
              where diff = y - x
-- Reinhard Zumkeller, Apr 25 2015

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]];
Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}, {i, 1, zz}];
u = Table[a[k], {k, 1, zz}]  (* Clark Kimberling, May 13 2015 *)

A175498_list, l, s, b1, b2 = [1,2], 2, 3, set(), set([1])
for n in range(3, 10**5):
    i = s
    while True:
        if not (i in b1 or i-l in b2):
            A175498_list.append(i)
            b1.add(i)
            b2.add(i-l)
            l = i
            while s in b1:
                b1.remove(s)
                s += 1
            break
        i += 1 # Chai Wah Wu, Dec 15 2014

More terms from Sean A. Irvine, Jan 27 2011

A257883 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 0.

Original entry on oeis.org

0, 1, 3, 2, 5, 9, 4, 10, 6, 11, 8, 15, 7, 16, 14, 22, 12, 23, 17, 27, 13, 25, 18, 31, 19, 33, 20, 35, 24, 40, 21, 38, 29, 47, 26, 45, 28, 48, 30, 51, 36, 58, 32, 55, 39, 63, 34, 59, 37, 64, 41, 67, 42, 70, 43, 72, 44, 74, 50, 81, 46, 78, 111, 49, 83, 52, 87

Offset: 1

Clark Kimberling, May 13 2015

Algorithm: 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). Let h be the least integer > -a(k) such that h is not in D(k) and 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 repeat inductively.
Conjecture: if a(1) is a 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.
Guide to related sequences:
a(1)   d(1)    (a(n))       (d(n))
0      0      A257883      A175499 except for initial terms
0      1      A257884      A175499
0      2      A257885      A257902
0      3      A257903      A257904
1      0      A175498      A175499 except for first term
1      1      A257905      A175499
2      0      A257908      A257909
2      1      A257910      A257909 except for initial terms
2      2      A257911      A257912

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

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

Cf. A081145, A175498, A257705.

a[1] = 0; 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]]
Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h,
   d[k + 1] = h, k = k + 1}, {i, 1, zz}];
u = Table[a[k], {k, 1, zz}]  (* A257883, = -1 + A175498 *)
Table[d[k], {k, 1, zz}] (* A175499 except that here first term is 0 *)

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

Also, A257883(n) = -1 + A175498(n) for n >= 1.

A257880 Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 0.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 13 2015

This is the sequence (d(n)) of differences associated with the sequence a = A257879.
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(1) = 2, d(1) = 0;
a(2) = 1, d(2) = -1;
a(3) = 3, d(3) = 2;
a(4) = 4, d(4) = 1.

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

Cf. A131389, A257705, A081145, A257883, A175499.

a[1] = 2; 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}] (* A257879 *)
Table[d[k], {k, 1, zz}]     (* A257880 *)

d(k) = a(k) - a(k-1) for k >=2, where a(k) = A257877(k).

A257884 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 1.

Original entry on oeis.org

0, 2, 1, 4, 8, 3, 9, 5, 10, 7, 14, 6, 15, 13, 21, 11, 22, 16, 26, 12, 24, 17, 30, 18, 32, 19, 34, 23, 39, 20, 37, 28, 46, 25, 44, 27, 47, 29, 50, 35, 57, 31, 54, 38, 62, 33, 58, 36, 63, 40, 66, 41, 69, 42, 71, 43, 73, 49, 80, 45, 77, 110, 48, 82, 51, 86, 52

Offset: 1

Clark Kimberling, May 13 2015

Algorithm:  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).  Let h be the least integer > -a(k) such that h is not in D(k) and 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 repeat inductively.
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 A257883 for a guide to related sequences.

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

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

Cf. A257883, A175498, A257705, A081145.

a[1] = 0; d[1] = 1; 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]]
Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h,
   d[k + 1] = h, k = k + 1}, {i, 1, zz}];
u = Table[a[k], {k, 1, zz}]  (* A257884 *)
Table[d[k], {k, 1, zz}]      (* A175499 *)

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

A257915 Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 3.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 12 2015

This is the sequence (d(n)) of differences associated with the sequence a = A257877.
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(1) = 0, d(1) = 3;
a(2) = 1, d(2) = 1;
a(3) = 3, d(3) = 2;
a(4) = 2, d(4) = -1.

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

Cf. A131389, A257705, A081145, A257883, A175499.

a[1] = 0; d[1] = 3; 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}]  (* A257877 *)
Table[d[k], {k, 1, zz}]  (* A257915 *)

d(k) = a(k) - a(k-1) for k >= 2, where a(k) = A257877(k).

A257918 Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 13 2015

This is the sequence (d(n)) of differences associated with the sequence a = A257882.
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(1) = 2, d(1) = 2;
a(2) = 1, d(2) = -1;
a(3) = 4, d(3) = 3;
a(4) = 5, d(4) = 1.

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

Cf. A131389, A257705, A081145, A257918, A175499.

a[1] = 2; d[1] = 2; 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}]  (* A257882 *)
Table[d[k], {k, 1, zz}]      (* A257918 *)

d(k) = a(k) - a(k-1) for k >=2, where a(k) = A257882(k).

A334404 a(0)=0; for n>0, a(n) is an integer not previously seen such that the sum of all previous terms plus a(n) equals the smallest prime number not yet created by any previous sum.

Original entry on oeis.org

0, 2, 1, 4, -2, 6, 8, -6, 10, 14, -20, 12, 18, -16, 22, -12, 20, -18, 16, 24, -10, 28, -34, 30, -26, 32, -24, 34, -4, 40, -60, 38, 36, -56, 44, -14, 42, -48, 26, 54, -72, 52, 48, -66, 50, -42, 46, -30, 60, -90, 62, -36, 58, -52, 64, -22, -8, 74, -38, 68, -54, 66, -78, 76, -70, 82, -46

Offset: 0

Scott R. Shannon, Sep 08 2020

sign

See A175499 for an equivalent sequence which sums to the smallest positive integer not yet created.

			a(1) = 2 as the sum of all previous terms plus a(1) = 0 + 2 = 2, where 2 has not previously occurred in the sequence and the prime 2 has not been previously created.
a(2) = 1 as the sum of all previous terms plus a(2) = 0 + 2 + 1 = 3, where 1 has not previously occurred in the sequence and the prime 3 has not been previously created.
a(3) = 4 as the sum of all previous terms plus a(3) = 0 + 2 + 1 + 4 = 7, where 4 has not previously occurred in the sequence and the prime 7 has not been previously created. Note that the next smallest uncreated prime after a(2) is 5 but that would require a(3) = 2 which is not allowed as a(1) = 2.
a(4) = -2 as the sum of all previous terms plus a(4) = 0 + 2 + 1 + 4 - 2 = 5, where -2 has not previously occurred in the sequence and the prime 5 has not been previously created.

Cf. A175499, A000040, A000217.

Nest[Block[{k = 1, s = Total[#[[All, 1]] ], i = 1, p}, While[Nand[FreeQ[#[[All, -1]], Set[p, Prime@ i]], FreeQ[#[[All, 1]], p - s] ], i++]; While[Nand[FreeQ[#[[All, 1]], k], k + s == p], If[k < 0, Set[k, -k + 1], k *= -1]]; Append[#, {k, p}]] &, {{0, 0}}, 66][[All, 1]] (* Michael De Vlieger, Sep 11 2020 *)

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A175498 a(1)=1. a(n) = the smallest positive integer not occurring earlier such that a(n)-a(n-1) doesn't equal a(k)-a(k-1) for any k with 2 <= k <= n-1.

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A257883 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 0.

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A257880 Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 0.

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A257884 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 1.

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A257915 Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 3.

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A257918 Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 2.

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A334404 a(0)=0; for n>0, a(n) is an integer not previously seen such that the sum of all previous terms plus a(n) equals the smallest prime number not yet created by any previous sum.

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