A354960 -id:A354960 - cp's OEIS frontend

A356430 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with the number of divisors of a(n-1).

1, 2, 4, 3, 6, 8, 10, 12, 9, 15, 14, 16, 5, 18, 20, 21, 22, 24, 26, 28, 27, 30, 32, 33, 34, 36, 39, 38, 40, 42, 44, 45, 46, 48, 25, 51, 50, 52, 54, 56, 58, 60, 57, 62, 64, 7, 66, 68, 63, 69, 70, 72, 74, 76, 75, 78, 80, 35, 82, 84, 81, 55, 86, 88, 90, 87, 92, 93, 94, 96, 98, 99, 100, 102, 104

Offset: 1

Scott R. Shannon, Aug 07 2022

nonn

The sequence is conjectured to be a permutation of the positive integers although it may take a large number of terms for the primes to appear, e.g., 19 has not occurred after 100000 terms. In the same range the only fixed points are 1, 2, and 9, and it is likely no more exist.

			a(9) = 9 as a(8) = 12 which has six divisors, and 9 is the smallest unused number that shares a factor with 6.

Scott R. Shannon, Image of the first 1000 terms. The green line is y = n.

Cf. A356431, A356432, A000005, A354960, A348086.

A356431 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with both a(n-1) and the number of divisors of a(n-1).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 3, 18, 9, 15, 20, 14, 16, 30, 22, 24, 26, 28, 21, 36, 27, 42, 32, 34, 38, 40, 44, 33, 48, 45, 39, 52, 46, 50, 54, 56, 58, 60, 51, 66, 62, 64, 70, 68, 72, 57, 76, 74, 78, 80, 5, 90, 63, 69, 84, 75, 81, 105, 96, 82, 86, 88, 92, 94, 98, 100, 102, 104, 106, 108, 87, 114, 110

Offset: 1

Scott R. Shannon, Aug 07 2022

nonn

The sequence is conjectured to be a permutation of the positive integers although it may take a large number of terms for the primes to appear, e.g., 17 has not occurred after 100000 terms. In the same range the only fixed points are the first two terms; it is likely no more exist although this is unknown.

			a(8) = 3 as a(7) = 12 which has six divisors, and 3 is the smallest unused number that shares a factor with both 12 and 6.

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

Cf. A356430, A356432, A000005, A354960, A348086.

A356432 a(1) = 1; for n > 1, a(n) is the smallest positive number not occurring earlier that shares a factor with a(n-1) plus the number of divisors of a(n-1).

Original entry on oeis.org

1, 2, 4, 7, 3, 5, 14, 6, 8, 9, 10, 12, 15, 19, 18, 16, 21, 20, 13, 24, 22, 26, 25, 28, 17, 38, 27, 31, 11, 39, 43, 30, 32, 34, 36, 33, 37, 42, 35, 45, 48, 29, 62, 40, 44, 46, 50, 49, 52, 54, 56, 58, 60, 51, 55, 59, 61, 57, 122, 63, 23, 65, 66, 64, 71, 73, 69, 146, 68, 70, 72, 74, 75, 78, 76, 41

Offset: 1

Scott R. Shannon, Aug 07 2022

nonn

The sequence is conjectured to be a permutation of the positive integers. In the first 250000 terms there are twenty-three fixed points: 1, 2, 12, 16, 27 ..., 2279, 5401, 7339. It is possibly no more exist although this is unknown.

			a(7) = 14 as a(6) = 5 which has two divisors, and 14 is the smallest unused number that shares a factor with 5 + 2 = 7.

Robert Israel, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 250000 terms. The green line is y = n.

Cf. A356430, A356431, A000005, A354960, A348086.

A[1]:= 1; S:= {$2..5000}:
for i from 2 do
  found:= false;
  t:= A[i-1] + numtheory:-tau(A[i-1]);
  for s in S do
    if igcd(s,t) > 1 then
      A[i]:= s;
      found:= true;
      S:= S minus {s};
      break
    fi
  od;
  if not found then break fi;
od:
seq(A[j],j=1..i-1); # Robert Israel, Jan 16 2023

A358082 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with Sum_{k=1..n-1} sigma(a(k)).

Original entry on oeis.org

1, 2, 4, 11, 23, 47, 5, 101, 7, 211, 3, 14, 22, 487, 6, 9, 8, 10, 1033, 12, 15, 13, 18, 16, 2203, 21, 46, 26, 29, 4583, 89, 9257, 20, 28, 35, 18661, 24, 17, 27, 37441, 30, 19, 25, 32, 33, 36, 34, 38, 39, 40, 42, 44, 45, 48, 37, 31, 50, 49, 52, 54, 56, 58, 60, 62, 63, 51, 57, 64, 55, 66, 69, 72

Offset: 1

Scott R. Shannon, Nov 02 2022

nonn

The sequence shows large jumps in value due to the sum occasionally forming a large prime, e.g., a(279) = 2650277753. The sequence is conjectured to be a permutation of the positive integers.

			a(7) = 5 as Sum_{k=1..6} sigma(a(k)) = Sum_{k=1..6} A000203(a(k)) = 1 + 3 + 7 + 12 + 24 + 48 = 95, and 5 is the smallest unused number that shares a factor with 95.

Cf. A000203, A358176, A358201, A064413, A356851, A356430, A354960.

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)).

Original entry on oeis.org

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

A356430 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with the number of divisors of a(n-1).

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A356431 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with both a(n-1) and the number of divisors of a(n-1).

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A356432 a(1) = 1; for n > 1, a(n) is the smallest positive number not occurring earlier that shares a factor with a(n-1) plus the number of divisors of a(n-1).

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Maple

A358082 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with Sum_{k=1..n-1} sigma(a(k)).

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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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Mathematica

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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