A337136 -id:A337136 - cp's OEIS frontend

A366473 a(1) = 1; thereafter a(n) = A337136(n) + A337136(n-1).

1, 3, 5, 8, 9, 10, 14, 15, 16, 19, 29, 48, 41, 53, 94, 67, 81, 82, 31, 47, 78, 60, 31, 80, 82, 42, 43, 64, 67, 158, 160, 51, 42, 44, 38, 39, 58, 62, 66, 67, 234, 237, 75, 74, 72, 75, 78, 82, 86, 90, 91, 94, 97, 145, 147, 101, 152, 153, 106, 110, 111, 112, 115, 81, 83, 143, 148, 129, 127, 190, 193, 259, 263, 333, 332, 137, 205, 212, 147, 149, 226, 223, 297

Offset: 1

N. J. A. Sloane, Oct 26 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, accentuating prime powers in red.

Cf. A337136.

nn = 120;
  c[_] := False;
  Array[Set[{a[#], c[#]}, {#, True}] &, 2]; Set[{i, j, u},
   Range[3]]; s = i + j;
  Do[k = u;
    While[Or[c[k], CoprimeQ[s, k]], k++];
    Set[{a[n], c[k], i, j, s}, {s, True, j, k, j + k}];
    If[k == u, While[c[u], u++]], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, Oct 26 2023 *)

A366474 Where n appears in A337136, or -1 if n never appears.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 7, 9, 10, 35, 13, 22, 16, 18, 19, 33, 23, 11, 25, 27, 26, 64, 29, 32, 31, 34, 36, 12, 37, 20, 38, 40, 39, 44, 42, 45, 46, 43, 47, 14, 48, 28, 49, 51, 50, 21, 53, 52, 55, 56, 58, 15, 59, 61, 60, 62, 63, 228, 65, 99, 24, 69, 68, 67, 71, 17, 76

Offset: 1

N. J. A. Sloane, Oct 26 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A337136.

nn = 120; c[_] := 0; Array[Set[{a[#], c[#]}, {#, #}] &, 2]; Set[{i, j, u}, Range[3]]; s = i + j;
Do[k = u; While[Or[c[k] > 0, CoprimeQ[s, k]], k++];
  Set[{a[n], c[k], i, j, s}, {k, n, j, k, j + k}];
  If[k == u, While[c[u] > 0, u++]], {n, 3, nn}];
TakeWhile[Array[c, nn], # > 0 &] (* Michael De Vlieger, Oct 26 2023 *)

A351001 a(0) = 0, a(1) = 1; for n > 1, a(n) is the smallest positive number which has not appeared which has a common factor with a(n-2) + a(n-1) but does not equal a(n-2) + a(n-1).

Original entry on oeis.org

0, 1, 2, 6, 4, 5, 3, 10, 26, 8, 12, 14, 13, 9, 11, 15, 16, 62, 18, 20, 19, 21, 22, 86, 24, 25, 7, 28, 30, 29, 118, 27, 35, 31, 32, 33, 39, 34, 146, 36, 38, 37, 40, 42, 41, 166, 23, 45, 17, 44, 122, 46, 48, 47, 50, 194, 52, 51, 206, 514, 54, 56, 55, 57, 49, 53, 58, 60, 59, 63, 61, 64

Offset: 0

Scott R. Shannon, Jan 28 2022

nonn

This is a permutation of the natural numbers. Up to 500000 terms the fixed points are 0, 1, 2, 4, 5, 15, 16, 18, 21, 22, 24, 25, 29, and it is likely no more exist.

			a(3) = 6 as a(1)+a(2) = 3, 6 does not equal 3, and gcd(3,6) > 1.
a(4) = 4 as a(2)+a(3) = 8, 4 does not equal 8, and gcd(8,4) > 1.

Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A337136, A064413, A336957, A098550, A089088, A251622.

s = {0, 1, 2}; u = 3; c[] = 0; Set[{i, j}, s[[-2 ;; -1]]]; Array[Set[c[s[[#]]], #] &, Length[s]]; s~Join~Reap[Do[Set[k, u]; While[Nand[c[k] == 0, GCD[i + j, k] > 1, i + j != k], k++]; Sow[k]; Set[c[k], n]; If[k == u, While[c[u] == 1, u++]]; i = j; j = k, {n, Length[s] + 1, 2^10}]][[-1, -1]] (* _Michael De Vlieger, Jan 28 2022 *)

A359799 a(1) = 1, a(2) = 3; for n > 2, a(n) is the smallest positive number which has not appeared that shares a factor with |a(n-1) - a(n-2)| while the difference |a(n) - a(n-1)| is distinct from all previous differences |a(i) - a(i-1)|, i=2..n-1.

Original entry on oeis.org

1, 3, 6, 12, 2, 10, 14, 26, 4, 11, 28, 17, 22, 35, 65, 5, 20, 36, 8, 32, 9, 23, 42, 76, 18, 38, 56, 15, 41, 16, 25, 54, 87, 21, 48, 27, 63, 24, 66, 7, 59, 13, 44, 93, 49, 84, 30, 62, 100, 19, 69, 106, 37, 90, 212, 34, 74, 122, 33, 89, 46, 129, 249, 39, 86, 141, 40, 101, 183, 50, 95, 159, 52

Offset: 1

Scott R. Shannon, Mar 07 2023

nonn

In the first 100000 terms the only fixed point is a(1) = 1; it is unknown if more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(5) = 2 as |a(4) - a(3)| = |12 - 6| = 6, and 2 is the smallest unused number that shares a factor with 6 while the difference |2 - a(4)| = |2 - 12| = 10 is distinct from all previous differences.

Scott R. Shannon, Image for n = 1..100000. The green line is a(n) = n.

Cf. A361314, A337136, A354755, A354727, A354687, A354753, A353989, A354087, A352763.

A350927 a(1)=1, a(2)=2; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1) * |a(n-1) - a(n-2)|, a(n)) > 1.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 8, 10, 5, 15, 12, 14, 7, 21, 16, 18, 20, 22, 11, 33, 24, 26, 13, 39, 27, 28, 30, 25, 35, 32, 34, 17, 51, 36, 38, 19, 57, 40, 42, 44, 46, 23, 69, 45, 48, 50, 52, 54, 56, 49, 63, 58, 29, 87, 60, 55, 65, 62, 31, 93, 64, 66, 68, 70, 72, 74, 37, 111, 75, 76, 78, 80, 82, 41, 123, 81

Offset: 1

Scott R. Shannon, Jan 28 2022

nonn

This is likely a permutation of the natural numbers. The sequence shows similar behavior to the EKG sequence A064413. In the first 500000 terms the fixed points are 1, 2, 77, 221, and it is likely no more exist.

			a(4) = 6 as a(3)*|a(3)-a(2)| = 4*2 = 8, 6 has not been used and gcd(6,8) > 1.
a(8) = 10 as a(7)*|a(7)-a(6)| = 8*1 = 8, 10 has not been used and gcd(10,8) > 1.

Scott R. Shannon, Image of the first 10000 terms. The green line is y = n.

Cf. A351001, A337136, A064413, A336957, A098550, A089088, A251622.

A353006 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with (a(n-1) + a(n-2))*|a(n-1) - a(n-2)|.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 7, 9, 10, 19, 12, 14, 13, 15, 16, 31, 18, 21, 24, 20, 11, 27, 22, 25, 30, 33, 28, 35, 36, 71, 40, 37, 39, 26, 45, 38, 42, 32, 34, 44, 46, 48, 47, 50, 51, 101, 52, 17, 23, 54, 49, 55, 56, 57, 113, 58, 60, 59, 63, 61, 62, 41, 66, 65, 131, 64, 67, 69, 68, 137, 70, 72, 74

Offset: 1

Scott R. Shannon, Apr 16 2022

nonn

The sequences is conjectured to be a permutation of the positive integers. In the first 500000 terms there are 637 fixed points: 1, 2, 3, 6, 9, ..., 180611, 189383, 298097. As the terms slowly move away from the line a(n) = n as n increases it is likely no more exist.

			a(4) = 5 as (a(3)+a(2))*|a(3)-a(2)| = (3+2)*|3-2| = 5, and 5 is the smallest unused number that shares a factor with 5.
a(5) = 4 as (a(4)+a(3))*|a(4)-a(3)| = (5+3)*|5-3| = 16, and 4 is the smallest unused number that shares a factor with 16.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n - 1..2^14, showing records in red, local minima in blue, fixed points highlighted in amber, and primes in green.
Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A337136, A352867, A352790, A350927.

nn = 74, c[] = 0; Array[Set[{a[#], c[#]}, {#, #}] &, 2]; u = 3; Do[m = (#1 + #2) Abs[(#1 - #2)] & @@ {a[i - 1], a[i - 2]}; k = u; While[Nand[c[k] == 0, ! CoprimeQ[m, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, Length[s] + 1, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 17 2022 *)

A360209 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n > 2, a(n) shares a factor with a(n-2) + a(n-1) but shares no factor with a(n-2).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 15, 7, 8, 9, 17, 10, 12, 11, 23, 14, 37, 27, 16, 43, 59, 18, 21, 13, 20, 22, 33, 25, 26, 24, 35, 295, 32, 36, 51, 29, 28, 19, 47, 30, 44, 259, 39, 34, 73, 107, 38, 40, 45, 119, 41, 46, 42, 55, 97, 48, 50, 49, 57, 52, 109, 63, 54, 65, 77, 56, 76, 69, 75, 58, 91, 149, 60, 66

Offset: 1

Scott R. Shannon, Jan 29 2023

nonn

To ensure the sequence is infinite another criterion must be satisfied when choosing a(n), namely a(n) + a(n-1) must contain a factor not in a(n-1). If this were not the case, a(n+1) = a(n) + a(n-1) would share a factor with both a(n) + a(n-1) and a(n-1), terminating the sequence.

In the first 100000 terms the fixed points for n > 2 are 3, 6, 441, 1677, 3629, 9701, 17131, although it is likely more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(7) = 15 as a(5) + a(6) = 4 + 6 = 10, and 15 is the smallest positive unused number that shares a factor with 10 but not with a(5) = 4.
a(41) = 44 as a(39) + a(40) = 47 + 30 = 77, and 44 shares a factor with 77 but not with a(39) = 47. Note that 42 also satisfies these criteria but 30 + 42 = 72 which shares all its factors with a(40) = 30, thus setting a(41) = 42 would make it impossible to find a(42).

Scott R. Shannon, Image for n=1..100000. The green line is a(n) = n.

Cf. A251604 (does not share with a(n-1)), A098550, A336957, A337136, A359557, A353239.

a(6) and above corrected by Scott R. Shannon, Mar 17 2023

A361314 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number which has not appeared that shares a factor with a(n-2) + a(n-1) while the sum a(n) + a(n-1) is distinct from all previous sums a(i) + a(i-1), i=2..n-1.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 7, 9, 10, 19, 29, 12, 41, 53, 14, 67, 15, 16, 31, 47, 13, 20, 18, 22, 24, 26, 25, 17, 27, 28, 11, 21, 36, 30, 32, 38, 34, 39, 73, 35, 33, 42, 45, 48, 51, 44, 40, 46, 43, 89, 50, 139, 49, 52, 101, 54, 55, 109, 56, 57, 113, 58, 60, 59, 63, 61, 62, 66, 64, 65, 69, 68, 137, 70

Offset: 1

Scott R. Shannon, Mar 08 2023

nonn

In the first 100000 terms the fixed points are 1, 2, 3, 6, 9, 10, 39, 91, 112; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(23) = 20 as a(21) + a(22) = 47 + 13 = 60, and 20 is the smallest unused number that shares a factor with 60 while the sum a(22) + 20 = 13 + 20 = 33 is distinct from all previous sums. Note that 18 is unused and shares a factor with 60 but the sum a(22) + 18 = 13 + 18 = 31 is the same as a(18) + a(19) = 15 + 16 = 31. This is the first term that differs from A337136.

Scott R. Shannon, Image for n = 1..100000. The green line is a(n) = n.

Cf. A337136, A359799, A354755, A354727, A354687, A354753, A353989, A354087, A352763.

A353082 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that is a divisor of (a(n-1) + a(n-2))*|a(n-1) - a(n-2)|.

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 13, 8, 7, 15, 11, 26, 37, 21, 16, 185, 39, 14, 25, 33, 29, 31, 6, 925, 19, 12, 217, 41, 22, 57, 35, 23, 24, 47, 71, 48, 17, 65, 32, 97, 43, 10, 53, 63, 20, 83, 103, 30, 73, 4429, 18, 401, 383, 28, 137, 55, 64, 51, 115, 128, 27, 101, 74, 45, 119, 82, 67, 149, 36, 113, 77

Offset: 1

Scott R. Shannon, Apr 22 2022

The sequence is finite, the final term being a(160)=143. See the examples. The maximum value is a(110)=35941 while the fixed points are 1, 2, 3, 8, 11, 67.

			a(6) = 9 as (a(5)+a(4))*|a(5)-a(4)| = (4+5)*|4-5| = 9, and 9 is the smallest unused number that is a divisor of 9. This is the first term that differs from A353006.
a(159) = 144, a(160) = 143. Therefore a(161) would be the smallest unused number that divides (143+144)*|143-144| = 287. The divisors are 1, 7, 41, 287, all of which have already appeared, so the sequence terminates.

Cf. A353006, A337136, A352867, A352790, A350927.

A353905 a(1) = 1; for n > 1, a(n) = smallest positive number that has not appeared that has a common factor with a(n-1) + the smallest unseen positive number.

Original entry on oeis.org

1, 3, 5, 7, 6, 2, 4, 8, 17, 10, 19, 12, 9, 14, 15, 13, 16, 18, 29, 20, 31, 21, 22, 11, 24, 47, 25, 26, 28, 27, 30, 53, 32, 33, 34, 36, 59, 38, 61, 35, 40, 39, 42, 45, 44, 67, 46, 23, 48, 50, 51, 52, 89, 49, 43, 54, 56, 57, 58, 55, 60, 97, 62, 63, 64, 101, 66, 103, 65, 68, 69, 70, 107, 72, 109

Offset: 1

Scott R. Shannon, May 10 2022

nonn

The sequence is conjectured to be a permutation of the positive integers. In the first 100000 terms the number 18869 holds the record for the greatest number of terms for which it is the lowest unseen number, 4769 terms in all. In the same range there are fifteen fixed points, the last being a(1204), and it is likely no more exist.

			a(2) = 3 as a(1) = 1, the smallest unseen positive number is 2, and 1 + 2 = 3, and 3 is the smallest number that has not yet appeared that shares a factor with 3.
a(5) = 6 as a(4) = 7, the smallest unseen positive number is 2, and 7 + 2 = 9, and 6 is the smallest number that has not yet appeared that shares a factor with 9.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^14, showing records in red and local minima in blue. Primes and fixed points are highlighted in green and gold, respectively.
Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Cf. A337136, A089088, A064413, A085229, A093714.

nn = 120; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, ! CoprimeQ[#, k]], k++] &[a[i - 1] + u]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* Michael De Vlieger, May 15 2022 *)

cp's OEIS Frontend

A366473 a(1) = 1; thereafter a(n) = A337136(n) + A337136(n-1).

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A366474 Where n appears in A337136, or -1 if n never appears.

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A351001 a(0) = 0, a(1) = 1; for n > 1, a(n) is the smallest positive number which has not appeared which has a common factor with a(n-2) + a(n-1) but does not equal a(n-2) + a(n-1).

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A359799 a(1) = 1, a(2) = 3; for n > 2, a(n) is the smallest positive number which has not appeared that shares a factor with |a(n-1) - a(n-2)| while the difference |a(n) - a(n-1)| is distinct from all previous differences |a(i) - a(i-1)|, i=2..n-1.

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A350927 a(1)=1, a(2)=2; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1) * |a(n-1) - a(n-2)|, a(n)) > 1.

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A353006 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with (a(n-1) + a(n-2))*|a(n-1) - a(n-2)|.

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A360209 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n > 2, a(n) shares a factor with a(n-2) + a(n-1) but shares no factor with a(n-2).

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A361314 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number which has not appeared that shares a factor with a(n-2) + a(n-1) while the sum a(n) + a(n-1) is distinct from all previous sums a(i) + a(i-1), i=2..n-1.

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A353082 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that is a divisor of (a(n-1) + a(n-2))*|a(n-1) - a(n-2)|.

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A353905 a(1) = 1; for n > 1, a(n) = smallest positive number that has not appeared that has a common factor with a(n-1) + the smallest unseen positive number.

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