A377041 First term of the n-th differences of the squarefree numbers. Inverse zero-based binomial transform of A005117.

1, 1, 0, 1, -3, 6, -8, 3, 22, -92, 252, -578, 1189, -2255, 3991, -6617, 10245, -14626, 18666, -19635, 12104, 13090, -69122, 171478, -332718, 552138, -798629, 982514, -901485, 116219, 2351842, -8715135, 23856206, -57926011, 130281064, -273804584, 535390333

Offset: 0

Author

Gus Wiseman, Oct 18 2024

Keywords

• sign

Crossrefs

The version for primes is A007442, noncomposites A030016, composites A377036.

This is the first column of A377038.

For nonsquarefree numbers we have A377049.

For prime-powers we have A377054.

A000040 lists the primes, differences A001223, seconds A036263.

A005117 lists the squarefree numbers, complement A013929 (differences A078147).

A073576 counts integer partitions into squarefree numbers, factorizations A050320.

A377042 gives first position of 0 in each row of A377038.

Cf. A053797, A053806, A061398, A072284, A076259, A120992, A376311, A376590, A376591, A377040, A377046.

Programs

Mathematica

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

Formula

The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:

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