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A377049 First term of the n-th differences of the nonsquarefree numbers. Inverse zero-based binomial transform of A013929.

%I A377049 #10 Nov 28 2024 11:28:30
%S A377049 4,4,-3,5,-6,4,3,-15,25,-10,-84,369,-1067,2610,-5824,12246,-24622,
%T A377049 47577,-88233,155962,-259086,393455,-512281,456609,191219,-2396571,
%U A377049 8213890,-21761143,50923029,-110269263,225991429,-444168664,844390152,-1561482492,2817844569
%N A377049 First term of the n-th differences of the nonsquarefree numbers. Inverse zero-based binomial transform of A013929.
%H A377049 Harvey P. Dale, <a href="/A377049/b377049.txt">Table of n, a(n) for n = 0..400</a>
%F A377049 The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:
%F A377049   p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)
%t A377049 nn=20;
%t A377049 Table[First[Differences[NestList[NestWhile[#+1&, #+1,SquareFreeQ[#]&]&,4,2*nn],k]],{k,0,nn}]
%t A377049 With[{nsf=Select[Range[1000],!SquareFreeQ[#]&]},Table[Differences[nsf,n],{n,0,40}]][[;;,1]] (* _Harvey P. Dale_, Nov 28 2024 *)
%Y A377049 The version for primes is A007442, noncomposites A030016, composites A377036.
%Y A377049 For squarefree instead of nonsquarefree numbers we have A377041.
%Y A377049 For antidiagonal-sums we have A377047, absolute A377048.
%Y A377049 For first position of 0 in each row we have A377050.
%Y A377049 A000040 lists the primes, differences A001223, seconds A036263.
%Y A377049 A005117 lists the squarefree numbers.
%Y A377049 A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
%Y A377049 A073576 counts integer partitions into squarefree numbers, factorizations A050320.
%Y A377049 Cf. A007674, A053797, A053806, A061398, A072284, A075526, A084758, A112925, A120992, A251092, A376311, A376591.
%K A377049 sign
%O A377049 0,1
%A A377049 _Gus Wiseman_, Oct 19 2024