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

1,2

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.

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.

Example

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.

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]

Crossrefs

Cf. A007947, A064413, A357963.

Sequence in context: A352187 A357963 A357994 * A255348 A342366 A336946

Adjacent sequences: A357939 A357940 A357941 * A357943 A357944 A357945

Keyword

nonn

Author

Michael De Vlieger, Oct 22 2022

Status

approved