A357942 a(1)=1, a(2)=2. Thereafter, if there are prime divisors p | a(n-1) that are coprime to a(n-2), a(n) is the least novel multiple of the product of these primes. Otherwise a(n) is the least novel multiple of the squarefree kernel of a(n-1). See comments.

1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 25, 30, 36, 42, 28, 56, 70, 35, 105, 27, 33, 11, 22, 26, 13, 39, 45, 40, 32, 34, 17, 51, 48, 38, 19, 57, 54, 44, 55, 50, 46, 23, 69, 60, 80, 90, 63, 49, 77, 66, 72, 78, 52, 104, 130, 65, 195, 75, 120

Offset: 1

Author

Michael De Vlieger, Oct 22 2022

Keywords

• nonn

Comments

Let k be the greatest common squarefree divisor of a(n-2) and a(n-1) and let s = A007947(a(n-1)). If k = 1, then a(n) = m_s*s, else a(n) = m_k*k, where m_i is the smallest multiple of i such that m*i does not appear in a(1..n-1).

Variant of A357963; a(21) = 36, but A357963(21) = 22.

Examples

			a(1) = 1, a(2) = 2 and 2 divides 2 but does not divide 1. Since 2 is the only prime divisor of 2, a(3) = 4, the smallest unused multiple of 2.
Since every prime divisor of a(3)=4 also divides a(2) = 2, a(4) = 6, the least novel multiple of the squarefree kernel of 4.
a(19,20) = (25,30); 2|30 and 3|30 but 2 and 3 do not divide 25. The smallest multiple of 2*3 = 6 not already in the sequence is 36. Therefore a(21) = 36.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Scatterplot of a(n), n = 1..2^20.
• Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^14, showing records in red, local minima in blue, highlighting primes in green and other prime powers in gold.

Crossrefs

Cf. A007947, A064413, A357963.

Programs

Mathematica

nn = 67; c[] = False; q[] = 1; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; q[2] = 2; Do[m = FactorInteger[a[n - 1]][[All, 1]]; If[Length@ # == 0, While[Set[k, #* q[#]]; c[k], q[#]++] &[Times @@ m], While[Set[k, #* q[#]]; c[k], q[#]++] &[Times @@ #]] &@ Select[m, CoprimeQ[#, a[n - 2]] &]; Set[{a[n], c[k]}, {k, True}], {n, 3, nn}]; Array[a, nn]