A091538 - OEIS

%I #20 Aug 29 2019 17:27:16

%S 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,

%T 0,17,15,27,40,72,96,128,0,19,21,28,54,80,144,192,256,0,23,22,30,56,

%U 108,160,288,384,512,0,29,25,42,60,112,216,320,576,768,1024

%N Triangle built from m-primes as columns.

%C 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.

%C The number N>=1 appears in column nr. m = A001222(N).

%H W. Lang, <a href="/A091538/a091538.txt">First 11 rows</a>.

%F 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.

%e From _Michael De Vlieger_, May 24 2017: (Start)

%e Chart a(n,m) read by antidiagonals:

%e n | m ->

%e ------------------------------------------------

%e 0 |    1     0     0     0     0     0     0 ... (A000007)

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

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

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

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

%e 5 |   32    48    72    80   108   112   120     (A014614)

%e 6 |   64    96   144   160   216   224   240     (A046306)

%e 7 |  128   192   288   320   432   448   480     (A046308)

%e 8 |  256   384   576   640   864   896   960     (A046310)

%e      ...

%e Triangle begins:

%e 0 |    1

%e 1 |    0    2

%e 2 |    0    3    4

%e 3 |    0    5    6    8

%e 4 |    0    7    9   12   16

%e 5 |    0   11   10   18   24   32

%e 6 |    0   13   14   20   36   48    64

%e 7 |    0   17   15   27   40   72    96   128

%e 8 |    0   19   21   28   54   80   144   192   256

%e      ...

%e (End)

%t 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 *)

%t 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 *)

%Y 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.

%Y A078840 is this table with the zeros omitted.

%K nonn,easy,tabl

%O 0,3

%A _Wolfdieter Lang_, Feb 13 2004