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Antidiagonal-sums of the absolute value of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).
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%I #6 Oct 19 2024 08:32:26

%S 4,8,10,12,14,18,21,28,34,40,47,74,96,110,138,286,715,2393,8200,25731,

%T 72468,184716,431575,934511,1892267,3605315,6494464,11116110,18134549,

%U 28348908,42701927,62290660,88313069,120999433,159769475,221775851,483797879

%N Antidiagonal-sums of the absolute value of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

%e The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 12.

%t q=Select[Range[120],CompositeQ];

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

%t Total/@Table[Abs[t[[j,i-j+1]]],{i,Length[q]/2},{j,i}]

%Y The version for prime instead of composite is A376681, absolute version of A140119.

%Y The version for noncomposite is A376684, absolute version of A376683.

%Y This is the antidiagonal-sums of absolute value of the array A377033.

%Y For squarefree instead of composite we have A377040, absolute version of A377039.

%Y For nonsquarefree instead of composite we have A377048, absolute version of A377047.

%Y For prime-power instead of composite we have A377053, absolute version of A377052.

%Y Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).

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

%Y A002808 lists the composite numbers, differences A073783, seconds A073445.

%Y A008578 lists the noncomposites, differences A075526.

%Y Cf. A018252, A065310, A065890, A333254, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377036.

%K nonn,new

%O 1,1

%A _Gus Wiseman_, Oct 18 2024