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A126279
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Triangle read by rows: T(k,n) is number of numbers <= 2^n that are products of k primes.
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7
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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; internal format)
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OFFSET
| 1,2
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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.
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LINKS
| Jonathan Vos Post & Robert G. Wilson v, Table of n, a(n) for n = 1..1276
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).
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EXAMPLE
| Triangle begins:
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
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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 Weisstein (eww(AT)wolfram.com) Feb 07 2006 *)
Table[ AlmostPrimePi[m, 2^n], {n, 16}, {m, n}] // Flatten
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CROSSREFS
| First column: A007053, 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
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KEYWORD
| tabl,less,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com) & Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 22 2006
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