A377036 First term of the n-th differences of the composite numbers. Inverse zero-based binomial transform of A002808.

4, 2, 0, -1, 2, -2, 0, 4, -8, 8, 0, -16, 32, -32, -1, 78, -233, 687, -2363, 8160, -25670, 72352, -184451, 430937, -933087, 1888690, -3597221, 6479696, -11086920, 18096128, -28307626, 42644791, -62031001, 86466285, -110902034, 110907489, -52325, -483682930

Offset: 0

Author

Gus Wiseman, Oct 18 2024

Keywords

• sign

Crossrefs

The version for prime instead of composite is A007442.

For noncomposite numbers we have A030016.

This is the first column (n=1) of A377033.

For row-sums we have A377034, absolute version A377035.

First zero positions are A377037, cf. A376678, A376855, A377042, A377050, A377055.

For squarefree instead of composite we have A377041, nonsquarefree A377049.

For prime-power instead of composite we have A377054.

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.

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

A008578 lists the noncomposites, differences A075526.

Cf: A018252, A065310, A065890, A140119, A173390, A333214, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

Programs

Mathematica

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

Formula

The inverse zero-based binomial transform of a sequence (q(0), q(1), ..., q(m)) is the sequence p given by:

p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)