A037088 Triangle read by rows: T(n,k) is number of numbers x, 2^n <= x < 2^(n+1), with k prime factors (counted with multiplicity).

2, 2, 2, 2, 4, 2, 5, 4, 5, 2, 7, 12, 6, 5, 2, 13, 20, 17, 7, 5, 2, 23, 40, 30, 20, 8, 5, 2, 43, 75, 65, 37, 21, 8, 5, 2, 75, 147, 131, 81, 41, 22, 8, 5, 2, 137, 285, 257, 173, 91, 44, 22, 8, 5, 2, 255, 535, 536, 344, 199, 96, 46, 22, 8, 5, 2, 464, 1062, 1033, 736, 403, 215, 99, 47

Offset

1,1

Comments

Sequence A092097 gives the limiting behavior at the end of the rows. - T. D. Noe, Feb 22 2008

Links

T. D. Noe, Rows n=1..24 of triangle, flattened

Example

The triangular array begins 2; 2,2; 2,4,2; 5,4,5,2; 7,12,6,5,2; ...
a(7) = 5 because the 3-almost primes between 16 and 32 are (18,20,27,28,30).

Mathematica

t[n_, k_] := Count[Range[2^n, 2^(n+1)-1], x_ /; Total[FactorInteger[x][[All, 2]]] == k]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 07 2013 *)

Crossrefs

A001222 counts factors of n. A000040, A001358, A014612-A014614 are special cases. A036378 and A025488 are applications of binary order A029837. Leading diagonal is essentially A036378 and has partial sums A007053.

Sequence in context: A089818 A067025 A049047 * A260722 A153436 A354359

Adjacent sequences: A037085 A037086 A037087 * A037089 A037090 A037091

Keyword

nonn,tabl,nice

Author

Alford Arnold

Extensions

More terms from Naohiro Nomoto, Jun 18 2001

Status

approved