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A091538 Triangle built from m-primes as columns. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..65.

W. Lang, First 11 rows.

FORMULA

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.

EXAMPLE

From Michael De Vlieger, May 24 2017: (Start)

Chart a(n,m) read by antidiagonals:

n | m ->

------------------------------------------------

0 |    1     0     0     0     0     0     0 ... (A000007)

1 |    2     3     5     7    11    13    17     (A000040)

2 |    4     6     9    10    14    15    21     (A001358)

3 |    8    12    18    20    27    28    30     (A014612)

4 |   16    24    36    40    54    56    60     (A014613)

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)

MATHEMATICA

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 *)

CROSSREFS

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.

Sequence in context: A229032 A117909 A261094 * A013584 A307320 A137372

Adjacent sequences:  A091535 A091536 A091537 * A091539 A091540 A091541

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Feb 13 2004

STATUS

approved

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Last modified January 28 22:35 EST 2020. Contains 331328 sequences. (Running on oeis4.)