

A078841


Main diagonal of the table of kalmost primes (A078840): a(n) = (n+1)st integer that is an nalmost prime.


17



1, 3, 9, 20, 54, 112, 240, 648, 1344, 2816, 5760, 12800, 26624, 62208, 129024, 270336, 552960, 1114112, 2293760, 4915200, 9961472, 20447232, 47775744, 96468992, 198180864, 411041792, 830472192, 1698693120, 3422552064, 7046430720
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OFFSET

0,2


COMMENTS

A kalmost prime is a positive integer that has exactly k prime factors counted with multiplicity.


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..228.
Eric Weisstein's World of Mathematics, Almost Prime.


FORMULA

Conjecture: Lim as n>inf. of a(n+1)/a(n) = 2.  Robert G. Wilson v, Nov 13 2007


EXAMPLE

a(0) = 1 since one is the multiplicative identity,
a(1) = 2nd 1almost prime is the second prime number = A000040(2) = 3,
a(2) = 3rd 2almost prime = 3rd semiprime = A001358(3) = 9 = {3*3}.
a(3) = 4th 3almost prime = A014612(4) = 20 = {2*2*5}.
a(4) = 5th 4almost prime = A014613(5) = 54 = {2*3*3*3},
a(5) = 6th 5almost prime = A014614(6) = 112 = {2*2*2*2*7}, ....


MATHEMATICA

f[n_] := Plus @@ Last /@ FactorInteger@n; t = Table[{}, {40}]; Do[a = f[n]; AppendTo[ t[[a]], n]; t[[a]] = Take[t[[a]], 10], {n, 2, 148*10^8}]; Table[ t[[n, n + 1]], {n, 30}] (* Robert G. Wilson v, Feb 11 2006 *)
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 *)
AlmostPrime[k_, n_] := Block[{e = Floor[ Log[2, n] + k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a  b/2, {p, e, 0, 1}]; a + b/2]; AlmostPrime[1, 1] = 2; lst = {}; Do[ AppendTo[lst, AlmostPrime[n1, n]], {n, 30}]; lst (* Robert G. Wilson v, Nov 13 2007 *)


CROSSREFS

Cf. A078840, A078842, A078843, A078844, A078445, A078846, A101695.
Sequence in context: A026566 A147356 A147416 * A147387 A146267 A151420
Adjacent sequences: A078838 A078839 A078840 * A078842 A078843 A078844


KEYWORD

nonn


AUTHOR

Benoit Cloitre and Paul D. Hanna, Dec 10 2002


EXTENSIONS

a(14)a(29) from Robert G. Wilson v, Feb 11 2006


STATUS

approved



