A362889 a(n) = n for n <= 2. Let i = a(n-2), j = a(n-1), q = gpd(i*j) = prime(s), and k = product of all distinct primes < q which do not divide i*j. For n > 2 a(n) is the least novel multiple of either k (if k is not the empty product), or of prime(s+1) if it is.

1, 2, 3, 5, 4, 6, 10, 7, 9, 20, 14, 12, 15, 21, 8, 25, 18, 28, 30, 11, 35, 24, 22, 70, 27, 33, 140, 13, 66, 105, 26, 44, 210, 39, 55, 42, 52, 110, 63, 65, 88, 84, 40, 77, 36, 45, 49, 16, 60, 56, 99, 50, 91, 132, 75, 98, 121, 90, 112, 143, 120

Offset: 1

Author

David James Sycamore, May 08 2023

Keywords

• nonn

Comments

In other words, if rad(i*j) is a primorial number A002110(s) then there are no primes < q which do not divide i*j (k = empty product), and a(n) is the least novel multiple of the smallest prime > q. Otherwise there are distinct primes < q which do not divide i*j, and a(n) is the least novel multiple of their product.

a(n) is prime iff the radical of the product of the two previous terms is a primorial number which has not occurred previously; see Example.

The sequence grows rapidly due to occurrences of i*j which differ greatly from primorial numbers.

Conjectured to be a permutation of the positive integers (but primes are not in order, e.g. 41 precedes 37).

Examples

			a(1,2) = 1, 2;  i*j = 2 = A002110(1), so a(3) = prime(2) = 3.
a(2,3) = 2,3; i*j = 6 = A002110(2) so a(4) = prime(3) = 5.
a(3,4) = 3,5; and 2 is the only prime < 3 which does not divide 15, so a(5) = 4, the least novel multiple of 2.
a(4,5) = 5,4; and 3 is the only prime < 5 which does not divide rad(20) = 10, so a(6) = 6, the least novel multiple of 3.
a(17,18) = 18,28; and 5 is the only prime not dividing rad(504) = 42, thus a(19) = 30, the least novel multiple of 5.
a(18,19) = 28,30; and rad(28*30) = 210 = A002110(4), so a(20) = prime(5) = 11.
a(52,53) = 50,91; and rad(i*j) = 2*5*7*13 = 910 and 3,11 are the only primes  < 13 which do not divide 910 so a(54) is 132, the least novel multiple of 3*11 = 33.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..256, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
• Michael De Vlieger, Log log scatterplot of log_10 a(n), n = 1..2^16.
• Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2^11, with a color function conveying multiplicity, where black = 1, red = 2, orange = 3, etc. The strip of color on the bottom of the image shows red if a(n) is prime, gold if composite prime power, green if squarefree composite, and blue if neither squarefree nor prime powers.
• Michael De Vlieger, Notes regarding this sequence

Crossrefs

Cf. A002110, A007947.

Programs

Mathematica

mm = 3; nn = 2^15;
  c[] := False; m[] := 1;
  Array[Set[{a[#], c[#]}, {#, True}] &, mm]; m[2]++;
  i = a[mm - 1]; j = a[mm]; q = FactorInteger[i j][[-1, 1]];
  Do[t = PrimePi /@ FactorInteger[i j][[All, 1]]; q = Last[t];
    s = DeleteCases[Range[q], _?(MemberQ[t, #] &)];
    If[Length[s] == 0,
      p = Prime[q + 1]; While[c[p*m[p]], m[p]++],
      p = Times @@ Map[Prime, s]; While[c[p*m[p]], m[p]++]];
    k = p*m[p]; Set[{a[n], c[k], i, j}, {k, True, j, k}], {n, mm + 1, nn}];
Array[a, nn] (* Michael De Vlieger, May 08 2023 *)