A377049 - OEIS

%I #10 Nov 28 2024 11:28:30

%S 4,4,-3,5,-6,4,3,-15,25,-10,-84,369,-1067,2610,-5824,12246,-24622,

%T 47577,-88233,155962,-259086,393455,-512281,456609,191219,-2396571,

%U 8213890,-21761143,50923029,-110269263,225991429,-444168664,844390152,-1561482492,2817844569

%N First term of the n-th differences of the nonsquarefree numbers. Inverse zero-based binomial transform of A013929.

%H Harvey P. Dale, <a href="/A377049/b377049.txt">Table of n, a(n) for n = 0..400</a>

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

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

%t nn=20;

%t Table[First[Differences[NestList[NestWhile[#+1&, #+1,SquareFreeQ[#]&]&,4,2*nn],k]],{k,0,nn}]

%t With[{nsf=Select[Range[1000],!SquareFreeQ[#]&]},Table[Differences[nsf,n],{n,0,40}]][[;;,1]] (* _Harvey P. Dale_, Nov 28 2024 *)

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

%Y For squarefree instead of nonsquarefree numbers we have A377041.

%Y For antidiagonal-sums we have A377047, absolute A377048.

%Y For first position of 0 in each row we have A377050.

%Y A000040 lists the primes, differences A001223, seconds A036263.

%Y A005117 lists the squarefree numbers.

%Y A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.

%Y A073576 counts integer partitions into squarefree numbers, factorizations A050320.

%Y Cf. A007674, A053797, A053806, A061398, A072284, A075526, A084758, A112925, A120992, A251092, A376311, A376591.

%K sign

%O 0,1

%A _Gus Wiseman_, Oct 19 2024