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A078840 Table of n-almost-primes T(n,k) (n >= 0, k > 0), read by antidiagonals, starting at T(0,1)=1 followed by T(1,1)=2. 33
1, 2, 3, 4, 5, 6, 8, 7, 9, 12, 16, 11, 10, 18, 24, 32, 13, 14, 20, 36, 48, 64, 17, 15, 27, 40, 72, 96, 128, 19, 21, 28, 54, 80, 144, 192, 256, 23, 22, 30, 56, 108, 160, 288, 384, 512, 29, 25, 42, 60, 112, 216, 320, 576, 768, 1024, 31, 26, 44, 81, 120, 224, 432, 640, 1152 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

An n-almost-prime is a positive integer that has exactly n prime factors.

This sequence is a rearrangement of the natural numbers. - Robert G. Wilson v, Feb 11 2006.

Each antidiagonal begins with the n-th prime and ends with 2^n.

From Eric Desbiaux, Jun 27 2009: (Start)

(A001222 gives A078840)

A001221 gives the Table:

1

-    2    3    4    5    7    8    9   11 ... A000961

-    6   10   12   14   15   18   20   21 ... A007774

-   30   42   60   66   70   78   84   90 ... A033992

-  210  330  390  420  462  510  546  570 ... A033993

- 2310 2730 3570 3990 4290 4620 4830 5460 ... A051270

Antidiagonals begin with A000961 and end with A002110.

Diagonal is A073329 which is last term in n-th row of A048692.

(End)

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..10011 (corrected by Ivan Neretin).

Eric Weisstein's World of Mathematics, Almost Prime.

EXAMPLE

Table begins:

1

-  2  3   5   7  11  13  17  19  23  29 ...

-  4  6   9  10  14  15  21  22  25  26 ...

-  8 12  18  20  27  28  30  42  44  45 ...

- 16 24  36  40  54  56  60  81  84  88 ...

- 32 48  72  80 108 112 120 162 168 176 ...

- 64 96 144 160 216 224 240 324 336 352 ...

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 *)

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]; Table[ AlmostPrime[k, n - k + 1], {n, 11}, {k, n}] // Flatten (* Robert G. Wilson v *)

mx = 11; arr = NestList[Take[Union@Flatten@Outer[Times, #, primes], mx] &, primes = Prime@Range@mx, mx]; Prepend[Flatten@Table[arr[[k, n - k + 1]], {n, mx}, {k, n}], 1] (* Ivan Neretin, Apr 30 2016 *)

(* The next code skips the initial 1.)

width = 15; (seq = Table[

  Rest[NestList[1 + NestWhile[# + 1 &, #, ! PrimeOmega[#] == z &] &,

  2^z, width - z + 1]] - 1, {z, width}]) // TableForm

Flatten[Map[Reverse[Diagonal[Reverse[seq], -width + #]] &, Range[width]]]

(* Peter J. C. Moses, Jun 05 2019 *)

PROG

(PARI) T(n, k)=if(k<0, 0, s=1; while(sum(i=1, s, if(bigomega(i)-n, 0, 1))<k, s++); s)

CROSSREFS

Cf. A078840, A078841, A078842, A078843, A078844, A078445, A078846, A109636.

T(1, k)=A000040(k), T(2, k)=A001358(k), T(3, k)=A014612(k), T(4, k)=A014613(k), T(5, k)=A014614(k), T(6, k)=A046306(k), T(7, k)=A046308(k), T(8, k)=A046310(k), T(9, k)=A046312(k), T(10, k)=A046314(k).

T(11, k)=A069272(k), T(12, k)=A069273(k), T(13, k)=A069274(k), T(14, k)=A069275(k), T(15, k)=A069276(k), T(16, k)=A069277(k), T(17, k)=A069278(k), T(18, k)=A069279(k), T(19, k)=A069280(k), T(20, k)=A069281(k).

T(k, 1)=A000079(k), T(k, 2)=A007283(k), T(k, 3)=A116453(k), T(k, k)=A101695(k), T(k, k+1)=A078841(k).

A091538 is this sequence with zeros inserted, making a square array.

Sequence in context: A266196 A215501 A117333 * A185974 A129129 A114622

Adjacent sequences:  A078837 A078838 A078839 * A078841 A078842 A078843

KEYWORD

nonn,tabf,hear

AUTHOR

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

EXTENSIONS

Edited by Robert G. Wilson v, Feb 11 2006

STATUS

approved

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Last modified December 11 07:31 EST 2019. Contains 329914 sequences. (Running on oeis4.)