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A298674 Square matrix read by antidiagonals up. Matrix of Dirichlet series associated with Sum_{n<=X} MangoldtLambda(n) * MangoldtLambda(n+2).

%I A298674 #38 Feb 07 2020 20:49:01
%S A298674 1,1,2,1,-2,-1,1,2,2,3,1,-2,-1,-1,2,1,2,-1,3,2,-2,1,-2,2,-1,-3,-4,2,1,
%T A298674 2,-1,3,2,-2,2,4,1,-2,-1,-1,-3,2,2,0,-3,1,2,2,3,2,4,2,4,3,4,1,-2,-1,
%U A298674 -1,2,2,-5,0,-3,-4,2,1,2,-1,3,-3,-2,2,4,-3,-6,2,-3
%N A298674 Square matrix read by antidiagonals up. Matrix of Dirichlet series associated with Sum_{n<=X} MangoldtLambda(n) * MangoldtLambda(n+2).
%C A298674 For n > 1: Sum_{k>=1} T(n,k) = log(A014963(n))*log(A014963(n+2)).
%C A298674 Triangular submatrix of this matrix is A298824.
%C A298674 Row sums of A298824 are found in A298825.  A298825(n)/n = A298826(n). A298826 appears to be relevant to the heuristic for the twin prime conjecture.
%C A298674 By varying the prime gap "h" in the program it appears that prime gaps that are powers of "h" have the same row sums of the triangular submatrix, which in turn seems to imply that prime gaps equal to powers of "h" have the same density.
%H A298674 Terence Tao, <a href="https://terrytao.wordpress.com/2017/07/06/correlations-of-the-von-mangoldt-and-higher-divisor-functions-i-long-shift-ranges/">Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges</a>. See second formula.
%F A298674 Let h = 2.
%F A298674 Let A(n,k) = 1 if n mod k = 0, otherwise 0.
%F A298674 Let B(n,k) = A008683(n)*n if k mod n = 0, otherwise 0.
%F A298674 Let T = A.B (where "." is matrix multiplication).
%F A298674 Take the Dirichlet convolution of a row in T(n,k) and a row in T(n+h,k) for n=1,2,3,4,5,... infinity, and form this matrix from the first columns of the convolutions. See Mathematica program for more precise description.
%e A298674 The square matrix starts:
%e A298674 {
%e A298674   {1,  2, -1,  3,  2, -2,  2,  4, -3,  4,  2, -3},
%e A298674   {1, -2,  2, -1,  2, -4,  2,  0,  3, -4,  2, -2},
%e A298674   {1,  2, -1,  3, -3, -2,  2,  4, -3, -6,  2, -3},
%e A298674   {1, -2, -1, -1,  2,  2,  2,  0, -3, -4,  2,  1},
%e A298674   {1,  2,  2,  3, -3,  4, -5,  4,  3, -6,  2,  6},
%e A298674   {1, -2, -1, -1,  2,  2,  2,  0, -3, -4,  2,  1},
%e A298674   {1,  2, -1,  3,  2, -2, -5,  4, -3,  4,  2, -3},
%e A298674   {1, -2,  2, -1, -3, -4,  2,  0,  3,  6,  2, -2},
%e A298674   {1,  2, -1,  3,  2, -2,  2,  4, -3,  4, -9, -3},
%e A298674   {1, -2, -1, -1, -3,  2,  2,  0, -3,  6,  2,  1}
%e A298674 }
%t A298674 h = 2; nn = 14;
%t A298674 A = Table[If[Mod[n, k] == 0, 1, 0], {n, nn}, {k, nn}];
%t A298674 B = Table[If[Mod[k, n] == 0, MoebiusMu[n]*n, 0], {n, nn}, {k, nn}];
%t A298674 T = (A.B);
%t A298674 TableForm[TwinMangoldt = Table[a = T[[All, kk]];
%t A298674     F1 = Table[If[Mod[n, k] == 0, a[[n/k]], 0], {n, nn}, {k, nn}];
%t A298674     b = T[[All, kk + h]];
%t A298674     F2 = Table[If[Mod[n, k] == 0, b[[n/k]], 0], {n, nn}, {k, nn}];
%t A298674     (F1.F2)[[All, 1]], {kk, nn - h}]];
%t A298674 Flatten[Table[TwinMangoldt[[n - k + 1, k]], {n, nn - h}, {k, n}]]
%Y A298674 Cf. A191898, A298824, A001694, A298825, A298826.
%K A298674 sign,tabl
%O A298674 1,3
%A A298674 _Mats Granvik_, Jan 24 2018