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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). 3
 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, 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, -1, 2, 2, -5, 0, -3, -4, 2, 1, 2, -1, 3, -3, -2, 2, 4, -3, -6, 2, -3 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For n > 1: Sum_{k>=1} T(n,k) = log(A014963(n))*log(A014963(n+2)). Triangular submatrix of this matrix is A298824. 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. 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. LINKS Terence Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. See second formula. FORMULA Let h = 2. Let A(n,k) = 1 if n mod k = 0, otherwise 0. Let B(n,k) = A008683(n)*n if k mod n = 0, otherwise 0. Let T = A.B (where "." is matrix multiplication). 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. EXAMPLE The square matrix starts: {   {1,  2, -1,  3,  2, -2,  2,  4, -3,  4,  2, -3},   {1, -2,  2, -1,  2, -4,  2,  0,  3, -4,  2, -2},   {1,  2, -1,  3, -3, -2,  2,  4, -3, -6,  2, -3},   {1, -2, -1, -1,  2,  2,  2,  0, -3, -4,  2,  1},   {1,  2,  2,  3, -3,  4, -5,  4,  3, -6,  2,  6},   {1, -2, -1, -1,  2,  2,  2,  0, -3, -4,  2,  1},   {1,  2, -1,  3,  2, -2, -5,  4, -3,  4,  2, -3},   {1, -2,  2, -1, -3, -4,  2,  0,  3,  6,  2, -2},   {1,  2, -1,  3,  2, -2,  2,  4, -3,  4, -9, -3},   {1, -2, -1, -1, -3,  2,  2,  0, -3,  6,  2,  1} } MATHEMATICA h = 2; nn = 14; A = Table[If[Mod[n, k] == 0, 1, 0], {n, nn}, {k, nn}]; B = Table[If[Mod[k, n] == 0, MoebiusMu[n]*n, 0], {n, nn}, {k, nn}]; T = (A.B); TableForm[TwinMangoldt = Table[a = T[[All, kk]];     F1 = Table[If[Mod[n, k] == 0, a[[n/k]], 0], {n, nn}, {k, nn}];     b = T[[All, kk + h]];     F2 = Table[If[Mod[n, k] == 0, b[[n/k]], 0], {n, nn}, {k, nn}];     (F1.F2)[[All, 1]], {kk, nn - h}]]; Flatten[Table[TwinMangoldt[[n - k + 1, k]], {n, nn - h}, {k, n}]] CROSSREFS Cf. A191898, A298824, A001694, A298825, A298826. Sequence in context: A205107 A228667 A336005 * A211354 A211352 A087187 Adjacent sequences:  A298671 A298672 A298673 * A298675 A298676 A298677 KEYWORD sign,tabl AUTHOR Mats Granvik, Jan 24 2018 STATUS approved

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Last modified July 30 07:49 EDT 2021. Contains 346348 sequences. (Running on oeis4.)