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

Author

Alford Arnold

Keywords

• nonn
• tabl
• nice

Comments

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

Examples

			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).
		

Links

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

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.

Programs

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 *)

Extensions

More terms from Naohiro Nomoto, Jun 18 2001