A084296 Triangle read by rows: T(n,k) = omega(prime(n)#+k) where prime(n)# is the n-th primorial (A002110) and omega is the number of distinct prime factors (A001221), 0 <= k < n.

1, 2, 1, 3, 1, 1, 4, 1, 2, 2, 5, 1, 2, 2, 3, 6, 2, 2, 3, 2, 2, 7, 3, 2, 3, 3, 2, 4, 8, 2, 3, 2, 4, 2, 3, 2, 9, 2, 3, 3, 3, 2, 4, 3, 4, 10, 3, 3, 2, 2, 2, 4, 3, 3, 2, 11, 1, 4, 3, 2, 4, 5, 4, 3, 3, 4, 12, 3, 3, 4, 2, 3, 6, 2, 3, 5, 4, 3, 13, 3, 4, 2, 3, 3, 3, 3, 3, 3, 6, 2, 4, 14, 2, 3, 2, 4, 5, 4, 5, 3, 3, 6, 4

Offset

1,2

Links

Table of n, a(n) for n=1..103.

Formula

T(n,k) = A001221(A002110(n)+k).

T(n,0) = n.

Example

Triangle begins:
  1;
  2, 1;
  3, 1, 1;
  4, 1, 2, 2;
  5, 1, 2, 2, 3;
  6, 2, 2, 3, 2, 2;
  7, 3, 2, 3, 3, 2, 4;
  ...

Mathematica

lf[x_] := Length[FactorInteger[x]]; q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]]; Flatten[Table[Table[lf[q[n]+j], {j, 0, n-1}], {n, 1, 20}], 1]

Crossrefs

Cf. A001221, A002110, A068069, A045983.

Sequence in context: A178340 A395221 A173261 * A209235 A062534 A143349

Adjacent sequences: A084293 A084294 A084295 * A084297 A084298 A084299

Keyword

nonn,tabl

Author

Labos Elemer, May 27 2003

Extensions

Revised by Sean A. Irvine, Mar 24 2026

Status

approved