A354803 -id:A354803 - cp's OEIS frontend

A354804 The products of consecutive terms in A354803.

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.

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

A354749 a(1) = 1; for n > 1, a(n) is the smallest positive number greater than 1 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, 2, 3, 4, 5, 2, 7, 3, 5, 6, 7, 4, 9, 2, 11, 3, 8, 5, 7, 8, 9, 5, 11, 4, 13, 2, 17, 3, 13, 5, 12, 7, 9, 10, 7, 11, 6, 13, 7, 15, 8, 11, 9, 13, 8, 17, 4, 19, 2, 23, 3, 16, 5, 17, 6, 19, 3, 25, 2, 27, 4, 23, 5, 19, 7, 16, 9, 14, 11, 10, 13, 11, 12, 13, 14, 15, 11, 16, 13, 15, 16, 17, 7, 20, 9

Offset: 1

Scott R. Shannon, Jun 06 2022

This is a variation of A354803 where all terms beyond the first must be greater than 1. In the first 1000000 terms the fixed points are 1,2,3,4,5,7, it is likely no more exist, while the smallest number not to have appeared is 2090. The sequence is conjectured to be a permutation of the positive integers.

See A354759 for the products of all pairs of terms.

			a(6) = 2 as a(5) = 5 and 2 is the smallest number greater than 1 that is coprime to 5 and whose product with 5, 2 * 5 = 10, has not previously appeared.

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

Cf. A354759, A354803, A354804, A354753, A354754, A088177.

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.

A354759 The products of pairs of consecutive terms in A354749.

Original entry on oeis.org

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.

A354858 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 both the sum a(n) + a(n-1) is distinct from all previous sums, a(i) + a(i-1), i=2..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, 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, 63, 7, 56, 6, 58, 10, 55, 20, 52, 22, 55, 25, 60, 9, 69, 12, 70, 14, 72, 15, 75, 18, 70, 21, 75, 24, 68, 26, 72, 28, 74, 30, 65

Offset: 1

Scott R. Shannon, Jun 09 2022

This sequence uses a combination of the term selection rules of A354755 and A354753. The first forty-five terms are the same as A354755 beyond which they differ; see the examples below. In the first 500000 terms only six terms are prime, 2,3,7,19, with 2 and 3 occurring twice, the last being a(47) = 7. It is unknown if more appear. The only fixed points are 1,2,4,6, and it is likely no more exist.

			a(7) = 3 as a(6) = 6, and 3 is the smallest number that shares a factor with 6 and whose sum and product with the previous term, 6 + 3 = 9 and 6 * 3 = 18, have not previously 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.
a(46) = 63 as a(45) = 3, and 63 is the smallest number that shares a factor with 3 and whose sum and product with the previous term, 3 + 63 = 66 and 3 * 63 = 189, have not previously appeared. Note 60 shares a factor with 3 but the product 3 * 60 = 180 has already occurred with a(19) * a(20) = 12 * 15 = 180, so 60 cannot be chosen. This is the first term to differ from A354755.

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

Cf. A354755, A354753, A088177, A064413, A354803.

lista(nn) = my(va = vector(nn), vp = vector(nn-2), 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]), vp) || #select(x->(x==k+va[n-1]), vs), k++); va[n] = k; vp[n-2] = k*va[n-1]; vs[n-2] = k+va[n-1];); va; \\ Michel Marcus, Jun 17 2022

cp's OEIS Frontend

A354804 The products of consecutive terms in A354803.

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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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A354749 a(1) = 1; for n > 1, a(n) is the smallest positive number greater than 1 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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A354759 The products of pairs of consecutive terms in A354749.

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A354858 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 both the sum a(n) + a(n-1) is distinct from all previous sums, a(i) + a(i-1), i=2..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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