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

0,1

Links

Table of n, a(n) for n=0..37.

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)

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}]

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.

Sequence in context: A093486 A385567 A259618 * A354681 A115143 A093556

Adjacent sequences: A377033 A377034 A377035 * A377037 A377038 A377039

Keyword

sign

Author

Gus Wiseman, Oct 18 2024

Status

approved