A361606 -id:A361606 - cp's OEIS frontend

A362754 a(1) = 1, a(2) = 6; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and also contains as a factor the smallest prime that is not a factor of a(n-1).

1, 6, 10, 12, 15, 18, 20, 24, 30, 14, 21, 28, 36, 40, 42, 35, 50, 45, 48, 60, 56, 54, 70, 63, 66, 55, 22, 33, 44, 72, 75, 78, 65, 26, 39, 52, 84, 80, 90, 98, 96, 100, 102, 85, 34, 51, 68, 108, 105, 110, 99, 88, 114, 95, 38, 57, 76, 120, 112, 126, 130, 117, 104, 132, 135, 138, 115, 46, 69, 92, 144

Offset: 1

Scott R. Shannon, May 02 2023

nonn

No term can be a prime power as each term must contain at least two distinct prime factors. This make the sequence similar to A360519 and A361606. A close examination of the lines of concentrated terms, see the attached images, shows they have a slight downward curvature. In the first 250000 terms the only fixed points are 1, 69, 87, 116825, although it is possible more exist for very large values of n.

			a(3) = 10 as a(2) = 6 = 2*3, and 10 is the smallest unused number that shares a factor with 6 while also containing 5 as a prime factor, the smallest prime not a factor of 6.
a(4) = 12 as a(3) = 10 = 2*5, and 12 is the smallest unused number that shares a factor with 10 while also containing 3 as a prime factor, the smallest prime not a factor of 10.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing squarefree composites in green, numbers neither squarefree nor prime powers in blue, and numbers in A286708 in large light blue.
Scott R. Shannon, Image of the first 250000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 250000 terms in color. Terms with a lowest prime factor 2, 3, 5, 7, 11, >=13 are colored white, red, yellow, green, blue, violet and light gray respectively.

Cf. A337687, A351495, A064413, A360519, A361606.

nn = 120; c[] := False; m[] := 2; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 6}]; j = a[2]; Do[q = 2; While[Divisible[j, q], q = NextPrime[q]]; k = m[q]; While[Or[c[#], PrimePowerQ[#], CoprimeQ[j, k]] &[q k], k++]; k *= q; While[c[m[q] q], m[q]++]; Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, May 02 2023 *)

A362600 a(1) = 1, a(2) = 6, a(3) = 10; for n > 3, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and a(n-2) and also contains as factors the smallest primes that are not factors of both a(n-1) and a(n-2).

Original entry on oeis.org

1, 6, 10, 15, 12, 20, 30, 42, 35, 40, 60, 84, 70, 45, 18, 50, 75, 24, 80, 90, 105, 14, 36, 120, 140, 21, 48, 150, 210, 154, 33, 54, 110, 135, 66, 100, 165, 72, 130, 180, 126, 175, 160, 168, 195, 170, 78, 225, 190, 96, 240, 280, 63, 102, 270, 315, 28, 108, 300, 350, 147, 114, 330, 420, 77, 22

Offset: 1

Scott R. Shannon, May 02 2023

nonn

No term can be a prime power as each term must contain at least two distinct prime factors. This make the sequence similar to A362754, A360519 and A361606. Some small composite numbers take many terms to appear, e.g., a(354476) = 65. Such terms are usually preceded by a term that contains all the lower primes as factors. In the first 500000 terms, other than the first term, there are no fixed points, and it is unknown if any exist.

			a(4) = 15 as a(2) = 6 = 2*3 and a(3) = 10 = 2*5, and 15 is the smallest unused number that shares a factor with 6 and 10 while also containing 5 and 3 as prime factors, the smallest primes not factors of 6 and 10 respectively. This is the first term to differ from A362754.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing squarefree numbers in green and nonsquarefree numbers in blue, highlighting nonsquarefree numbers that are powerful (in A001694) with large light blue dots.
Scott R. Shannon, Image of the first 250000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 250000 terms in color. Terms with a lowest prime factor 2, 3, 5, 7, 11, >=13 are colored white, red, yellow, green, blue, violet and light gray respectively.

Cf. A362754, A337687, A351495, A064413, A360519, A361606, A053669.

nn = 120; c[_] := False;
f[x_] := If[OddQ[x], 2, y = 3; While[Divisible[x, y], y = NextPrime[y]]; y];
MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {1, 6, 10}];
i = a[2]; j = a[3]; q = 5; u = 12;
Do[qq = f[j]; k = Ceiling[u/#] &[q*qq];
  While[Or[c[#], CoprimeQ[i, #], CoprimeQ[i, j]] &[k*q*qq], k++];
  k *= q*qq;
  Set[{a[n], c[k], i, j, q}, {k, True, j, k, qq}];
  If[k == u, While[Or[c[u], PrimePowerQ[u]], u++]], {n, 4, nn}];
Array[a, nn] (* Michael De Vlieger, May 09 2023 *)

cp's OEIS Frontend

A362754 a(1) = 1, a(2) = 6; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and also contains as a factor the smallest prime that is not a factor of a(n-1).

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