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A091538
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Triangle built from m-primes as columns.
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7
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1, 0, 2, 0, 3, 4, 0, 5, 6, 8, 0, 7, 9, 12, 16, 0, 11, 10, 18, 24, 32, 0, 13, 14, 20, 36, 48, 64, 0, 17, 15, 27, 40, 72, 96, 128, 0, 19, 21, 28, 54, 80, 144, 192, 256, 0, 23, 22, 30, 56, 108, 160, 288, 384, 512, 0, 29, 25, 42, 60, 112, 216, 320, 576, 768, 1024
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OFFSET
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0,3
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COMMENTS
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m-primes (also called m-almost primes) are the numbers which have precisely m prime factors counting multiple factors. 1 is included as 0-prime.
The number N>=1 appears in column nr. m = A001222(N).
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LINKS
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FORMULA
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For n>=m>=1: a(n, m)= (n-m+1)-th member in the strictly monotonically increasing sequence of numbers N satisfying: N=product(p(k)^(e_k), k=1..) with p(k) := A000040(k) (k-th prime) such that sum(e_k, k=1..) = m, where the e_k are nonnegative. if m=0 : a(n, 0)=1 if n=0 else 0. If n<m then a(n, m)=0.
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EXAMPLE
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Chart a(n,m) read by antidiagonals:
n | m ->
------------------------------------------------
5 | 32 48 72 80 108 112 120 (A014614)
6 | 64 96 144 160 216 224 240 (A046306)
7 | 128 192 288 320 432 448 480 (A046308)
8 | 256 384 576 640 864 896 960 (A046310)
...
Triangle begins:
0 | 1
1 | 0 2
2 | 0 3 4
3 | 0 5 6 8
4 | 0 7 9 12 16
5 | 0 11 10 18 24 32
6 | 0 13 14 20 36 48 64
7 | 0 17 15 27 40 72 96 128
8 | 0 19 21 28 54 80 144 192 256
...
(End)
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MATHEMATICA
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With[{nn = 11}, Function[s, Function[t, Table[Function[m, If[m == 1, Boole[k == 1], t[[m, k]]]][n - k + 1], {n, nn}, {k, n, 1, -1}]]@ Map[Position[s, #][[All, 1]] &, Range[0, nn]]]@ PrimeOmega@ Range[2^nn]] (* or *)
a = {1}; Do[Block[{r = {Prime@ n}}, Do[AppendTo[r, SelectFirst[ Range[a[[-(n - i)]] + 1, 2^n], PrimeOmega@ # == i &]], {i, 2, n - 1}]; a = Join[a, {0}, If[n == 1, {}, r], {2^n}]], {n, 11}]; a (* Michael De Vlieger, May 24 2017 *)
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CROSSREFS
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The column sequences (without leading zeros) are: A000007, A000040 (primes), A001358, A014612-4, A046306, A046308, A046310, A046312, A046314, A069272-A069281 for m=0..20, respectively.
A078840 is this table with the zeros omitted.
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KEYWORD
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AUTHOR
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STATUS
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approved
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