a(n,k)  tabf head (staircase) for  A116864
  
  Product of pure prime parts partitions of n in Abramowitz-Stegun order.
  The entry is 0 iff a partition has a part 1 or a prime part.

  
  n\k   1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
 
  1     0  

  2     2  0

  3     3  0  0  

  4     0  0  4  0  0  

  5     5  0  6  0  0  0  0

  6     0  0  0  9  0  0  8  0  0  0  0 

  7     7  0 10  0  0  0  0 12  0  0  0  0  0  0  0 
 
  8     0  0  0 15  0  0  0  0  0 18  0  0  0  0 16  0  0  0  0  0  0  0  
 
  9     0  0 14  0  0  0  0  0  0 20  0 27  0  0  0  0  0 24  0  0  0  0  0  0  0  0  0  0  0  0 

 10     0  0  0 21  0 25  0  0  0  0  0 30  0  0  0  0  0  0  0  0  0  0 36  0  0  0  0  0  0 32  0  0  0  0  0  0  0  0  0  0  0  0

   .
   .
   .

  n\k   1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42


 The sequence of row lengths is A000041: [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...] (partition numbers).

 One could add the row for n=0 with a 0, if the part 0 is considered for n=0, and only for this n.

 For the ordering of this tabf array a(n,m) see Abramowitz-Stegun ref. pp. 831-2. 
 
 E.g. a(4,3) refers to the third partition of n=4 in this order, namely (2,2), which has only prime parts and  a(4,3)=2*2=4.
 
 E.g. a(4,4) refers to the fourth partition of n=4 in this ordering, namely (1^2,2)=(2,1,1), whence a(4,4)=0 because 

 it has a part 1. 
    
This array shows the `inverse' of the sequence A001414= [0,2,3,4,5,5,7,6,6,7,11,7,13,9,8,8,17,8,19,9,10,13,23,9,10,15,9,11,29,10,..].
the sum of prime factors of n>=1 (called sopfr(n)). 

E.g. 5=A001414(5)=A001414(6) corresponds to the two nonzero entries in row n=5, which relates to the partitons (5) and (3,2).

Simlarly row n=6 has two nonzero entries from the pure prime partitions (3,3) and (2,2,2), corresponding to 6=A001414(9)=A001414(6).

Therefore this array gives in row number n the values m for which A001414(m)=n>=2. E.g. n=10 appears 5 times in A001414, 
namely for k= 21,25,30 and 32.

The row sums give A002098(n+1) = [0,2,3,4,11,17,29,49,85,144,...], n>=1.

If all nonzero numbers are replaced by 1 the row sums give the number of partitions of n with only prime parts,  

A000607(n+1)= [0,1,1,1,2,2,3,3,4,5,,,,]. n>=1.



 ############################################### eof ##################################################################