A354749 -id:A354749 - cp's OEIS frontend

A354759 The products of pairs of consecutive terms in A354749.

2, 6, 12, 20, 10, 14, 21, 15, 30, 42, 28, 36, 18, 22, 33, 24, 40, 35, 56, 72, 45, 55, 44, 52, 26, 34, 51, 39, 65, 60, 84, 63, 90, 70, 77, 66, 78, 91, 105, 120, 88, 99, 117, 104, 136, 68, 76, 38, 46, 69, 48, 80, 85, 102, 114, 57, 75, 50, 54, 108, 92, 115, 95, 133, 112, 144, 126, 154, 110, 130

Offset: 1

Scott R. Shannon, Jun 06 2022

nonn

See A354749 for further details.

			a(3) = 12 as A354749(3) * A354749(4) = 3 * 4 = 12.

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

Cf. A354749, A354803, A354804, A354753, A354754, A088177.

A354753 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 product a(n) * a(n-1) is distinct from all previous products a(i) * a(i-1), i=2..n-1.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 2, 10, 4, 8, 6, 3, 3, 9, 6, 6, 10, 5, 5, 15, 3, 21, 6, 12, 8, 8, 10, 10, 12, 9, 9, 15, 6, 14, 2, 22, 4, 14, 7, 7, 21, 9, 18, 8, 14, 10, 15, 12, 14, 14, 16, 8, 20, 10, 22, 6, 26, 2, 34, 4, 26, 8, 22, 11, 11, 33, 3, 39, 6, 32, 8, 30, 9, 24, 12, 21, 14, 20, 15, 15, 21, 18, 18

Offset: 1

Scott R. Shannon, Jun 06 2022

This sequence uses a similar rule to A088177 but here all neighboring terms also share a factor. In the first 500000 terms the fixed points are 1,2,4,6, it is likely no more exist, while the smallest number not to have appeared is 1153. The sequence is conjectured to be a permutation of the positive integers.

See A354754 for the products of all pairs of terms.

			a(7) = 2 as a(6) = 6 and 2 is the smallest positive number that shares a factor with 6 and whose product with 6, 2 * 6 = 12, has not previously appeared.

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, a(n) = 2 in blue, fixed points highlighted in gold.
Scott R. Shannon, Image of the first 50000 terms.

Cf. A354754, A354803, A354804, A354749, A354759, A088177, A064413.

nn = 120; c[] = 0; a[1] = c[1] = 1; a[2] = a[2] = j = 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), vp = 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]), vp), k++); va[n] = k; vp[n-2] = k*va[n-1];); va; \\ Michel Marcus, Jun 14 2022

A354803 a(1) = 1; for n > 1, a(n) is the smallest positive number that is coprime to a(n-1) and the product a(n) * a(n-1) is distinct from all previous products a(i) * a(i-1), i=2..n-1.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 3, 5, 1, 7, 2, 5, 4, 7, 3, 8, 1, 9, 2, 11, 1, 13, 2, 15, 4, 9, 5, 7, 6, 11, 3, 13, 4, 11, 5, 8, 7, 9, 8, 11, 7, 10, 9, 11, 10, 13, 5, 16, 1, 17, 2, 19, 1, 23, 2, 25, 1, 27, 2, 29, 1, 31, 2, 37, 1, 32, 3, 16, 7, 12, 11, 13, 6, 17, 3, 19, 4, 17, 5, 19, 6, 23, 3, 25, 4, 23

Offset: 1

Scott R. Shannon, Jun 07 2022

This sequences uses similar a similar rule to A088177 but here all neighboring terms are coprime. In the first 1000000 terms the only fixed point is the first term while the smallest number not to have appeared is 2054. The sequence is conjectured to be a permutation of the positive integers.

See A354804 for the products of all pairs of terms.

			a(5) = 1 as a(4) = 3 and 1 is the smallest positive number that is coprime to 3 and whose product with 3, 1 * 3 = 3, has not previously appeared.

Scott R. Shannon, Image of the first 500000 terms.

Cf. A354804, A354749, A354759, A354753, A354754, A088177, A317024.

A354754 The products of consecutive terms in A354753.

Original entry on oeis.org

2, 4, 8, 16, 24, 12, 20, 40, 32, 48, 18, 9, 27, 54, 36, 60, 50, 25, 75, 45, 63, 126, 72, 96, 64, 80, 100, 120, 108, 81, 135, 90, 84, 28, 44, 88, 56, 98, 49, 147, 189, 162, 144, 112, 140, 150, 180, 168, 196, 224, 128, 160, 200, 220, 132, 156, 52, 68, 136, 104, 208, 176, 242, 121, 363, 99

Offset: 1

Scott R. Shannon, Jun 06 2022

nonn

See A354753 for further details.

			a(6) = 12 as A354753(6) * A354753(7) = 6 * 2 = 12.

Scott R. Shannon, Image of the first 50000 terms. The green line is y = n.

Cf. A354753, A354803, A354804, A354749, A354759, A088177, A064413.

A354804 The products of consecutive terms in A354803.

Original entry on oeis.org

1, 2, 6, 3, 4, 12, 15, 5, 7, 14, 10, 20, 28, 21, 24, 8, 9, 18, 22, 11, 13, 26, 30, 60, 36, 45, 35, 42, 66, 33, 39, 52, 44, 55, 40, 56, 63, 72, 88, 77, 70, 90, 99, 110, 130, 65, 80, 16, 17, 34, 38, 19, 23, 46, 50, 25, 27, 54, 58, 29, 31, 62, 74, 37, 32, 96, 48, 112, 84, 132, 143, 78, 102

Offset: 1

Scott R. Shannon, Jun 07 2022

nonn

See A354803 for further details.

			a(6) = 12 as A354803(6) * A354803(7) = 4 * 3 = 12.

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

Cf. A354803, A354749, A354759, A354753, A354754, A088177.

cp's OEIS Frontend

A354759 The products of pairs of consecutive terms in A354749.

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A354753 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 product a(n) * a(n-1) is distinct from all previous products a(i) * a(i-1), i=2..n-1.

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A354803 a(1) = 1; for n > 1, a(n) is the smallest positive number that is coprime to a(n-1) and the product a(n) * a(n-1) is distinct from all previous products a(i) * a(i-1), i=2..n-1.

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A354754 The products of consecutive terms in A354753.

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A354804 The products of consecutive terms in A354803.

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