A358082 -id:A358082 - cp's OEIS frontend

A358176 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with sigma(a(n-1)).

1, 2, 3, 4, 7, 6, 8, 5, 9, 13, 10, 12, 14, 15, 16, 31, 18, 21, 20, 22, 24, 25, 62, 26, 27, 28, 30, 32, 33, 34, 36, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 49, 19, 54, 55, 56, 57, 58, 60, 63, 64, 127, 66, 68, 69, 70, 72, 65, 74, 75, 76, 77, 78, 80, 81, 11, 82, 84, 86, 87, 85, 88, 90, 91, 92

Offset: 1

Scott R. Shannon, Nov 02 2022

nonn

The sequence is conjectured to be a permutation of the positive integers. In the first 500000 terms the fixed points are 1, 2, 3, 4, 6, 9, 12. It is unlikely more exist although this is unknown.

			a(8) = 5 as a(7) = 8 and sigma(8) = A000203(8) = 15, and 5 is the smallest unused number that shares a factor with 15.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 200000 terms. The green line is a(n) = n.
Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^14, showing records in red and local minima in blue, highlighting primes in green and other prime powers in gold. Extreme records and local minima are labeled in red and blue respectively, and composite prime powers in orange.

Cf. A000203, A358082, A358201, A064413, A356851, A356430, A354960.

nn = 120; c[] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = 3; Do[Set[{k, s}, {u, DivisorSigma[1, a[n - 1]]}]; While[Or[c[k], CoprimeQ[s, k]], k++]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Nov 05 2022 *)

A358201 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with sigma(max_{k=1..n-1}a(k)).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 8, 5, 9, 13, 10, 12, 14, 15, 16, 31, 18, 20, 22, 24, 26, 28, 30, 32, 21, 27, 33, 34, 36, 35, 39, 38, 40, 25, 42, 44, 45, 46, 48, 50, 51, 52, 49, 54, 55, 56, 57, 58, 60, 62, 63, 64, 127, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108

Offset: 1

Scott R. Shannon, Nov 03 2022

The sequence is conjectured to be a permutation of the positive integers. In the first 150000 terms the fixed points are 1, 2, 3, 4, 6, 9, 12, 93, 6003, 6881, 16269, 100707, 114839, 116999. It is likely more exist.

			a(9) = 9 as sigma(max_{k=1..8}a(k)) = sigma(8) = A000203(8) = 15, and 9 is the smallest unused number that shares a factor with 15.

Scott R. Shannon, Image of the first 150000 terms. The green line is a(n) = n.

Cf. A000203, A358176, A358082, A064413, A356851, A356430, A354960.

A358175 a(1) = 1, a(2) = 2; a(3) = 3; for n > 3, a(n) is the smallest positive number not occurring earlier that shares a factor with Sum_{k=1..n-1} A001065(a(k)), where A001065(m) is the sum of the proper divisors of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 19, 10, 7, 29, 9, 12, 14, 15, 18, 16, 20, 127, 22, 24, 26, 28, 21, 233, 13, 25, 241, 11, 27, 30, 32, 35, 33, 17, 34, 36, 433, 31, 39, 38, 40, 42, 44, 45, 48, 727, 46, 50, 797, 49, 52, 51, 54, 57, 37, 60, 56, 55, 58, 62, 63, 1259, 64, 66, 69, 68, 70, 65, 1579, 72, 78, 77, 74

Offset: 1

Scott R. Shannon, Nov 02 2022

nonn

The sequence is conjectured to be a permutation of the positive integers. In the first 10000 terms, other than the first three terms, there are thirty-five fixed points, the last being 2051. It is plausible no more exist although this is unknown.

			a(9) = 10 as Sum_{k=1..8} A001065(a(k)) = 0 + 1 + 1 + 3 + 1 + 6 + 7 + 1 = 20, and 10 is the smallest unused number that shares a factor with 20.

Cf. A001065, A027751, A358082, A064413, A356851, A356430, A354960.

cp's OEIS Frontend

A358176 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with sigma(a(n-1)).

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A358201 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with sigma(max_{k=1..n-1}a(k)).

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A358175 a(1) = 1, a(2) = 2; a(3) = 3; for n > 3, a(n) is the smallest positive number not occurring earlier that shares a factor with Sum_{k=1..n-1} A001065(a(k)), where A001065(m) is the sum of the proper divisors of m.

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