A126279 - OEIS

A126279

Triangle read by rows: T(k,n) is number of numbers <= 2^n that are products of k primes.

1, 2, 1, 4, 2, 1, 6, 6, 2, 1, 11, 10, 7, 2, 1, 18, 22, 13, 7, 2, 1, 31, 42, 30, 14, 7, 2, 1, 54, 82, 60, 34, 15, 7, 2, 1, 97, 157, 125, 71, 36, 15, 7, 2, 1, 172, 304, 256, 152, 77, 37, 15, 7, 2, 1, 309, 589, 513, 325, 168, 81, 37, 15, 7, 2, 1, 564, 1124, 1049, 669, 367, 177, 83, 37

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

Adolf Hildebrand, On the number of prime factors of an integer. Ramanujan revisited (Urbana-Champaign, Ill., 1987), 167 - 185, Academic Press, Boston, MA, 1988.

Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 205 - 211.

LINKS

Jonathan Vos Post & Robert G. Wilson v, Table of n, a(n) for n = 1..1330

Jonathan Vos Post & Robert G. Wilson v, Regular Triangle of A126279 for the first 52 rows, with some holes.

Eric Weisstein's World of Mathematics, Semiprime.

Eric Weisstein's World of Mathematics, Almost Prime.

EXAMPLE

Triangle begins:

2 1

4 2 1

6 6 2 1

11 10 7 2 1

18 22 13 7 2 1

31 42 30 14 7 2 1

54 82 60 34 15 7 2 1

97 157 125 71 36 15 7 2 1

172 304 256 152 77 37 15 7 2 1

MATHEMATICA

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[ PrimePi[ n/Times @@ Prime[ Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[ Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]] ]]]; (* Eric W. Weisstein, Feb 07 2006 *)

Table[ AlmostPrimePi[m, 2^n], {n, 16}, {m, n}] // Flatten

CROSSREFS

First column: A007053, second column: A125527, third column: A127396, 4th column: A334069. The last row reversed: A052130; the n-th row's sum: A000225 = 2^n -1.

Cf. A126280: same array but for powers of ten.

Sequence in context: A124840 A145118 A124927 * A135837 A027144 A158303

Adjacent sequences: A126276 A126277 A126278 * A126280 A126281 A126282

KEYWORD

tabl,less,nonn

AUTHOR

Jonathan Vos Post & Robert G. Wilson v, Dec 22 2006

STATUS

approved