A377033 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the composite numbers (A002808).

4, 6, 2, 8, 2, 0, 9, 1, -1, -1, 10, 1, 0, 1, 2, 12, 2, 1, 1, 0, -2, 14, 2, 0, -1, -2, -2, 0, 15, 1, -1, -1, 0, 2, 4, 4, 16, 1, 0, 1, 2, 2, 0, -4, -8, 18, 2, 1, 1, 0, -2, -4, -4, 0, 8, 20, 2, 0, -1, -2, -2, 0, 4, 8, 8, 0, 21, 1, -1, -1, 0, 2, 4, 4, 0, -8, -16, -16

Offset: 0

Author

Gus Wiseman, Oct 17 2024

Keywords

• sign
• tabl

Comments

Row n is the k-th differences of A002808 = the composite numbers.

Examples

			Array begins:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   4     6     8     9    10    12    14    15    16
  k=1:   2     2     1     1     2     2     1     1     2
  k=2:   0    -1     0     1     0    -1     0     1     0
  k=3:  -1     1     1    -1    -1     1     1    -1    -1
  k=4:   2     0    -2     0     2     0    -2     0     2
  k=5:  -2    -2     2     2    -2    -2     2     2    -2
  k=6:   0     4     0    -4     0     4     0    -4    -1
  k=7:   4    -4    -4     4     4    -4    -4     3    10
  k=8:  -8     0     8     0    -8     0     7     7   -29
  k=9:   8     8    -8    -8     8     7     0   -36    63
Triangle begins:
    4
    6    2
    8    2    0
    9    1   -1   -1
   10    1    0    1    2
   12    2    1    1    0   -2
   14    2    0   -1   -2   -2    0
   15    1   -1   -1    0    2    4    4
   16    1    0    1    2    2    0   -4   -8
   18    2    1    1    0   -2   -4   -4    0    8
   20    2    0   -1   -2   -2    0    4    8    8    0
   21    1   -1   -1    0    2    4    4    0   -8  -16  -16
		

Crossrefs

Initial rows: A002808, A073783, A073445.

The version for primes is A095195 or A376682.

A version for partitions is A175804, cf. A053445, A281425, A320590.

Triangle row-sums are A377034, absolute version A377035.

Column n = 1 is A377036, for primes A007442 or A030016.

First position of 0 in each row is A377037.

Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).

A000040 lists the primes, differences A001223, seconds A036263.

A008578 lists the noncomposites, differences A075526.

Cf. A065310, A065890, A084758, A173390, A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

Programs

Mathematica

nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,PrimeQ]&,4,2*nn],k],nn],{k,0,nn}]

Formula

A(i,j) = Sum_{k=0..j} (-1)^(j-k) binomial(j,k) A002808(i+k).