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A078841 Main diagonal of the table of k-almost primes (A078840): a(n) = (n+1)-st integer that is an n-almost 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A k-almost 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 1-almost prime is the second prime number = A000040(2) = 3,

a(2) = 3rd 2-almost prime = 3rd semiprime = A001358(3) = 9 = {3*3}.

a(3) = 4th 3-almost prime = A014612(4) = 20 = {2*2*5}.

a(4) = 5th 4-almost prime = A014613(5) = 54 = {2*3*3*3},

a(5) = 6th 5-almost 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[n-1, 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

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Last modified January 18 14:07 EST 2021. Contains 340254 sequences. (Running on oeis4.)