cp's OEIS Frontend

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

%I A377054 #5 Oct 22 2024 08:00:00
%S A377054 1,1,0,0,0,1,-5,15,-34,63,-97,115,-54,-251,1184,-3536,8736,-18993,
%T A377054 37009,-64545,98442,-121393,82008,147432,-860818,2710023,-7110594,
%U A377054 17077281,-38873146,85085287,-179965647,367885014,-725051280,1372311999,-2481473550,4257624252
%N A377054 First term of the n-th differences of the powers of primes. Inverse zero-based binomial transform of A000961.
%F A377054 The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:
%F A377054   p(j) = sum_{k=0..j} (-1)^(j-k)*binomial(j,k)*q(k)
%e A377054 The sixth differences of A000961 begin: -5, 10, -9, 1, 6, -10, 16, -18, ..., so a(6) = -5.
%t A377054 q=Select[Range[100],#==1||PrimePowerQ[#]&];
%t A377054 Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]/2}]
%Y A377054 The version for primes is A007442, noncomposites A030016, composites A377036.
%Y A377054 For squarefree numbers we have A377041, nonsquarefree A377049.
%Y A377054 This is the first column of the array A377051.
%Y A377054 For antidiagonal-sums we have A377052, absolute A377053.
%Y A377054 For positions of first zeros we have A377055.
%Y A377054 A000040 lists the primes, differences A001223, seconds A036263.
%Y A377054 A000961 lists the powers of primes, differences A057820.
%Y A377054 A001597 lists perfect-powers, complement A007916.
%Y A377054 A008578 lists the noncomposites, differences A075526.
%Y A377054 A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
%Y A377054 Cf. A025475, A053707, A093555, A174965, A175804, A246655, A361102, A376340, A376596, A377056.
%K A377054 sign
%O A377054 0,7
%A A377054 _Gus Wiseman_, Oct 22 2024