A377034 Antidiagonal-sums of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

4, 8, 10, 8, 14, 14, 11, 24, 10, 20, 37, -10, 56, 26, -52, 260, -659, 2393, -8128, 25703, -72318, 184486, -430901, 933125, -1888651, 3597261, -6479654, 11086964, -18096083, 28307672, -42644743, 62031050, -86466235, 110902085, -110907437, 52379, 483682985

Offset: 1

Author

Gus Wiseman, Oct 17 2024

Keywords

• sign

Comments

Row-sums of the triangle version of A377033.

Examples

			The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 8.
		

Crossrefs

The version for prime instead of composite is A140119, noncomposite A376683.

This is the antidiagonal-sums of the array A377033, absolute version A377035.

For squarefree instead of composite we have A377039, absolute version A377040.

For nonsquarefree instead of composite we have A377047, absolute version A377048.

For prime-power instead of composite we have A377052, absolute version A377053.

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

A000040 lists the primes, differences A001223, second A036263.

A002808 lists the composite numbers, differences A073783, second A073445.

A008578 lists the noncomposites, differences A075526.

Cf. A018252, A065310, A065890, A333254, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377036.

Programs

Mathematica

q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}];
Total/@Table[t[[j,i-j+1]],{i,Length[q]/2},{j,i}]