A091538 Triangle built from m-primes as columns.

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

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: A349339 A117909 A261094 * A340991 A013584 A307320

Adjacent sequences: A091535 A091536 A091537 * A091539 A091540 A091541

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Feb 13 2004

Status

approved