A354727 -id:A354727 - cp's OEIS frontend

A354739 First differences of A354727.

1, 2, 4, -2, -3, 6, 3, 8, -15, 5, 12, -11, 22, -18, 9, -10, -7, 14, 7, -12, 10, -13, 26, -21, 16, -17, 34, -24, 15, -6, -4, 18, -25, 20, -5, 24, -34, 25, -20, 21, -8, 27, -9, -28, -19, 38, 19, -32, 28, -26, -23, 46, 23, -40, 13, 30, -35, 33, -62, 31, 32, -47, 94, -87, 36, -27, -14, 35, -16, 17

Offset: 1

Scott R. Shannon, Jun 05 2022

sign

See A354727 for further details.

			a(5) = -3 as A354727(6) - A354727(5) = 3 - 6 = -3.

Scott R. Shannon, Image of the first 100000 terms. The blue line is y = 0.

Cf. A354727, A354687, A354575, A354679, A354688.

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.

A354575 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1) and the difference a(n) - a(n-1) is distinct from all previous differences.

Original entry on oeis.org

1, 2, 5, 3, 7, 4, 9, 8, 15, 11, 6, 17, 10, 19, 13, 21, 23, 12, 25, 16, 31, 14, 33, 20, 37, 18, 41, 26, 47, 22, 49, 27, 43, 29, 35, 53, 24, 55, 28, 57, 34, 59, 38, 71, 30, 67, 32, 73, 36, 79, 39, 61, 45, 77, 46, 81, 91, 40, 87, 44, 83, 50, 99, 52, 97, 42, 95, 51, 65, 89, 63, 101, 48, 103, 54, 113

Offset: 1

Scott R. Shannon, Jun 05 2022

This sequence uses a similar rule to A354688 but here the sign of the difference between a(n-1) and a(n) is considered. This leads to the terms showing much more erratic behavior than A354688; see the linked image.
In the first 200000 terms the fixed points are 1,2,8,35, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.
See A354679 for the differences between terms.

			a(9) = 15 as a(8) = 8, and 15 is the smallest unused number that is coprime to 8 and whose difference from the previous term, 15 - 8 = 7, has not appeared. Note that 11 and 13 are coprime to 8 but their differences from 8, namely 3 and 5, have already appeared as differences between previous pairs of terms.
a(15) = 13 as a(14) = 19, and 13 is the smallest unused number that is coprime to 19 and whose difference from the previous term, 13 - 19 = -6, has not appeared. Note that 12 is coprime to 19 and smaller than 13 but its difference from 19, namely -7, has already appeared as a difference between a(13) and a(12).

Scott R. Shannon, Image of the first 200000 terms. The green line is y = n.

Cf. A354679, A354688, A354727, A354739, A354687.

A354679 First differences of A354575.

Original entry on oeis.org

1, 3, -2, 4, -3, 5, -1, 7, -4, -5, 11, -7, 9, -6, 8, 2, -11, 13, -9, 15, -17, 19, -13, 17, -19, 23, -15, 21, -25, 27, -22, 16, -14, 6, 18, -29, 31, -27, 29, -23, 25, -21, 33, -41, 37, -35, 41, -37, 43, -40, 22, -16, 32, -31, 35, 10, -51, 47, -43, 39, -33, 49, -47, 45, -55, 53, -44, 14, 24, -26

Offset: 1

Scott R. Shannon, Jun 05 2022

See A354575 for further details.

			a(3) = -2 as A354575(4) - A354575(3) = 3 - 5 = -2.

Scott R. Shannon, Image of the first 200000 terms. The blue line is y = 0.

Cf. A354575, A354688, A354727, A354739, A354687.

A354755 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that shares a factor with a(n-1) and 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, 2, 4, 4, 6, 3, 9, 6, 8, 8, 10, 10, 12, 9, 15, 10, 16, 12, 15, 15, 18, 14, 20, 15, 21, 18, 20, 20, 22, 22, 24, 21, 27, 24, 26, 26, 28, 21, 35, 20, 38, 19, 57, 3, 60, 2, 62, 4, 64, 6, 63, 9, 66, 8, 70, 7, 77, 11, 88, 2, 78, 3, 84, 2, 80, 5, 60, 32, 62, 31, 93, 3, 99, 6, 92, 8, 96, 10, 85, 25

Offset: 1

Scott R. Shannon, Jun 06 2022

In the first 500000 terms the fixed points are 1,2,4,6,2388,2390,2392,2394; it is likely no more exist. In the same range many numbers do not appear, the lowest five being 59,67,73,89,97. It is possible these and many other numbers never appear although this is unknown.

			a(7) = 3 as a(6) = 6, and 3 is the smallest number that shares a factor with 6 and whose sum with the previous term, 6 + 3 = 9, has not appeared. Note 2 shares a factor with 6 but 6 + 2 = 8, and a sum of 8 has already occurred with a(4) + a(5) = 4 + 4 = 8, so 2 cannot be chosen.

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 a(n) = 2 in blue, highlighting fixed points in gold.
Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A064413, A354727, A354687, A354753, A353989, A354087, A352763

nn = 120; c[] = 0; a[1] = c[1] = 1; a[2] = j = 2; c[3] = 2; Do[k = 2; While[Nand[c[j + k] == 0, ! CoprimeQ[j, k]], k++]; Set[{a[n], c[j + k]}, {k, n}]; j = k, {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Jun 15 2022 *)

lista(nn) = my(va = vector(nn), vs = vector(nn-2)); va[1] = 1; va[2] = 2; for (n=3, nn, my(k=2); while ((gcd(k, va[n-1]) == 1) || #select(x->(x==k+va[n-1]), vs), k++); va[n] = k; vs[n-2] = k+va[n-1];); va; \\ Michel Marcus, Jun 15 2022

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.

cp's OEIS Frontend

A354739 First differences of A354727.

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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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A354575 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1) and the difference a(n) - a(n-1) is distinct from all previous differences.

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A354679 First differences of A354575.

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A354755 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that shares a factor with a(n-1) and 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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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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