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First term of the n-th differences of the composite numbers. Inverse zero-based binomial transform of the composite numbersA002808.

Cf. : A018252, A065310, A065890, `A140119, A173390, `A233671, `A258025, `A258026, `A333214, `A333254, `A350004, `A376656, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

Cf: A065310, A140119, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative).

A000040 lists the primes, differences A001223, second seconds A036263.

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

First column of the following (A377033):

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

t=Table[Sum[(-1)^(j-k)*Binomial[j, k]*q[[1+k]], {k, 0, j}], {j, 0, Length[q]/2-1}]

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

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

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

allocated for Gus Wiseman

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

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

0,1

First column of the following (A377033):

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

q=Select[Range[100], CompositeQ];

t=Table[Sum[(-1)^(j-k)*Binomial[j, k]*q[[1+k]], {k, 0, j}], {j, 0, Length[q]/2}]

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, second A036263.

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

A008578 lists the noncomposites, differences A075526.

Cf. A018252, A065890, `A173390, `A233671, `A258025, `A258026, `A333214, `A333254, `A350004, `A376656, A376680.

Cf: A065310, A140119, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative).

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Gus Wiseman, Oct 18 2024

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editing

allocated for Gus Wiseman

allocated

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