A367750 a(1,2) = 1,2. let i,j represent a(n-2), a(n-1) respectively and let M(i) = {i,j}. For n > 2, if one of i,j is prime (p), and the other (m) is not, then if p!|m, a(n) is the least novel multiple of p. If p|m, a(n) = p*m. If both terms in M are nonprime a(n) is the least novel multiple of the smallest prime which does not divide i*j.

1, 2, 4, 8, 3, 6, 18, 5, 10, 50, 9, 7, 14, 98, 12, 15, 21, 16, 20, 24, 28, 25, 27, 22, 30, 35, 11, 33, 363, 26, 40, 36, 42, 45, 44, 49, 39, 32, 55, 48, 56, 60, 66, 63, 65, 34, 51, 70, 77, 54, 75, 84, 88, 80, 57, 91, 38, 69, 85, 46, 72, 90, 105, 99, 52, 95, 78

Offset

1,2

Comments

From the definition it follows that both terms i,j in M cannot be prime, for if {i,j} is the first occurrence of two primes in M, then h, the term preceding i cannot be prime. With h nonprime and i prime we have either i|h or i!|h, and in either case j is a multiple of p, hence not prime.

Conjectured to be a permutation of the positive integers with the primes occurring in order.

Links

Table of n, a(n) for n=1..67.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue, highlighting numbers in the last category that are also squareful in light blue.

Example

M(1) = {1,2}; 2 prime and 2!|1-->a(3) = 4 (least novel multiple of 2).
M(2) = {2,4}, and 2|4--> a(4) = 2*4 = 8.
M(4) = {4,8}, no prime term --> a(5) = 3, least novel multiple of the smallest prime which does not divide 4*8 = 32.
M(8) = {8,3}, 3 prime!|8 -->a(6) = 6, least novel multiple of 3.
M(3) = {3,6}, 3|6--> a(7) = 3*6 = 18.
M(6) = {6,18} both terms nonprime--> a(8) = 5, least novel multiple of the smallest prime which does not divide 6*18 = 108.

Mathematica

nn = 120; c[_] := False; m[_] := 1;
q[x_] := Block[{qq}, qq = 2;
  While[Divisible[x, qq], qq = NextPrime[qq]]; qq];
Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2];
Set[{i, j}, {a[1], a[2]}];
Do[If[Xor[PrimeQ[i], PrimeQ[j]],
  Set[{k, p}, SortBy[{i, j}, PrimeQ]];
    If[Divisible[k, p],
      Set[k, i*j],
      While[c[Set[k, p m[p]]], m[p]++]],
    Set[k, q[i*j]]; While[c[k m[k]], m[k]++]; k *= m[k]];
   Set[{a[n], c[k], i, j}, {k, True, j, k}],
  {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Nov 29 2023 *)

Crossrefs

Sequence in context: A344534 A050170 A087089 * A197382 A332521 A246363

Adjacent sequences: A367747 A367748 A367749 * A367751 A367752 A367753

Keyword

nonn

Author

David James Sycamore, Nov 29 2023

Extensions

More terms from Michael De Vlieger, Nov 29 2023

Status

approved