A362754 -id:A362754 - cp's OEIS frontend

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).

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 *)

A363444 a(n) = n for n <= 3; for n > 3, a(n) is the smallest positive number that has not yet appeared that includes as factors the distinct prime factors of a(n-2) and a(n-1) that are not shared between a(n-2) and a(n-1).

Original entry on oeis.org

1, 2, 3, 6, 4, 9, 12, 8, 15, 30, 10, 18, 45, 20, 24, 60, 5, 36, 90, 25, 42, 210, 35, 48, 420, 70, 21, 120, 140, 63, 150, 280, 84, 75, 350, 126, 105, 40, 168, 315, 50, 252, 525, 80, 294, 630, 55, 462, 840, 110, 231, 1050, 220, 693, 1260, 330, 77, 1470, 660, 154, 735, 990, 308, 945, 1320, 616, 1155

Offset: 1

Scott R. Shannon, Jun 02 2023

nonn

The last prime to appear in the first 10000 terms is a(17) = 5, and it is unknown if more appear. The largest terms increase rapidly in size, e.g., a(8924) = 2233642178577810, although subsequent terms can be significantly smaller. It is unknown is all numbers eventually appear.

			a(4) = 6 as a(2) = 2 and a(3) = 3 contain the distinct prime factors 2 and 3 respectively, both of which only appear in one term. Therefore a(4) is the smallest unused number that contains both 2 and 3 as factors, which is 6.
a(6) = 9 as a(4) = 6 = 2*3 and a(5) = 4 = 2*2, so 3 is the only prime factor that is not shared between these terms. Therefore a(6) is the smallest unused number that contains 3 as a factor, which is 9.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^10, showing primes in red, composite prime powers in gold, squarefree composites in green, and other numbers in blue.
Michael De Vlieger, Plot p(i)^e(i) | a(n) at (x,y) = (n,i) for n = 1..2048, 12X vertical exaggeration, with a color function representing e(i), where black indicates e(i) = 1, red indicates e(i) = 2, yellow-green = 3, green = 4, and blue = 5. The bar at bottom indicates primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.

Cf. A098550, A336957, A362600, A362754, A351495, A359804.

nn = 120; c[] := False; m[] := 1; Array[Set[{a[#], c[#]}, {#, True}] &, 3];
i = {a[2]}; j = {a[3]}; Do[q = Times @@ SymmetricDifference[i, j]; While[c[Set[k, q m[q]]], m[q]++]; Set[{a[n], c[k], i, j}, {k, True, j, FactorInteger[k][[All, 1]]}], {n, 4, nn}]; Array[a, nn] (* Michael De Vlieger, Jun 05 2023 *)

cp's OEIS Frontend

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).

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