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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))

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