A377054 First term of the n-th differences of the powers of primes. Inverse zero-based binomial transform of A000961.

1, 1, 0, 0, 0, 1, -5, 15, -34, 63, -97, 115, -54, -251, 1184, -3536, 8736, -18993, 37009, -64545, 98442, -121393, 82008, 147432, -860818, 2710023, -7110594, 17077281, -38873146, 85085287, -179965647, 367885014, -725051280, 1372311999, -2481473550, 4257624252

Offset: 0

Author

Gus Wiseman, Oct 22 2024

Keywords

• sign

Examples

			The sixth differences of A000961 begin: -5, 10, -9, 1, 6, -10, 16, -18, ..., so a(6) = -5.
		

Crossrefs

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

For squarefree numbers we have A377041, nonsquarefree A377049.

This is the first column of the array A377051.

For antidiagonal-sums we have A377052, absolute A377053.

For positions of first zeros we have A377055.

A000040 lists the primes, differences A001223, seconds A036263.

A000961 lists the powers of primes, differences A057820.

A001597 lists perfect-powers, complement A007916.

A008578 lists the noncomposites, differences A075526.

A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.

Cf. A025475, A053707, A093555, A174965, A175804, A246655, A361102, A376340, A376596, A377056.

Programs

Mathematica

q=Select[Range[100],#==1||PrimePowerQ[#]&];
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)