A378722 a(1) = 1, a(2) = 2. For n > 2, a(n) is the smallest number not already a term which is divisible by A007947(a(n-1)/A020639(a(n-1))).

1, 2, 3, 4, 6, 9, 12, 18, 15, 5, 7, 8, 10, 20, 30, 45, 60, 90, 75, 25, 35, 14, 21, 28, 42, 63, 84, 126, 105, 70, 140, 210, 315, 420, 630, 525, 175, 245, 49, 56, 98, 77, 11, 13, 16, 22, 33, 44, 66, 99, 132, 198, 165, 55, 88, 110, 220, 330, 495, 660, 990, 825, 275

Offset

1,2

Comments

In other words a(n) is the least novel multiple of the squarefree kernel of a(n-1)/p, where p is the smallest prime factor of a(n-1).

If a(n-1) is prime, a(n) is the smallest unused number in the sequence so far.

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

Links

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..2^14, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple also shows powerful numbers that are not prime powers.

Example

a(2)= 2 (given), therefore a(3) is the smallest number not already a term which is divisible by the squarefree kernel of 2/2 = 1. Thus a(3) = 3. Likewise a(4) = 4. Since A007947(4/2) = 2, a(5) = 6, the least novel multiple of 2.

Mathematica

nn = 120; c[_] := False; m[_] := 1;
Do[Set[{a[i], c[i]}, {i, True}], {i, 2}]; j = a[2];
Do[k = Times @@ FactorInteger[j/FactorInteger[j][[1, 1]] ][[All, 1]];
  While[c[k*m[k]], m[k]++]; k *= m[k];
  Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Dec 06 2024 *)

Crossrefs

Cf. A007947, A020639.

Sequence in context: A062121 A094995 A375980 * A018591 A309591 A018669

Adjacent sequences: A378717 A378720 A378721 * A378723 A378729 A378730

Keyword

nonn,easy,new

Author

David James Sycamore, Dec 05 2024

Extensions

More terms from Michael De Vlieger, Dec 06 2024.

Status

approved