A298825 -id:A298825 - cp's OEIS frontend

A298826 a(n) = A298825(n)/n.

1, 0, 0, 1, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0, 4, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -2

Offset: 1

Mats Granvik, Jan 27 2018

sign

From Mats Granvik, Apr 06 2019: (Start)
Positions of nonzero entries appear to be given by A001694.
The limit lim_{k->oo} Sum_{n=1..k} a(n)/n appears to converge to some number.
Sum_{n=1..30000} a(n)/n = 1.31897... This appears to be in agreement with:
(1/30000)*Sum_{n<=30000} log(A014963(n))*log(A014963(n+2)) = 1.30351...
If the limit can be proven to converge to a number greater than 1, then it is true that Sum_{n<=X} log(A014963(n))*log(A014963(n+2)) > X, where ">" means "greater than" as in usual inequality notation.
The twin prime conjecture, according to Terence Tao on his blog, is that Sum_{n<=X} log(A014963(n))*log(A014963(n+2)) >> X where ">>" means "asymptotically greater than". There he also says that the first Hardy Littlewood conjecture states that Sum_{n<=X} log(A014963(n))*log(A014963(n+h)) = S(h)*X + o(X), where "S(h)" is the singular series.
Compare this to the prime number theorem which is lim_{X->oo} (Sum_{n<=X} log(A014963(n)))/X = 1.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..2500
Mats Granvik, Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture.
Terence Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. See second formula.

Cf. A298674, A298824, A298825, A001694, A361430.

nn = 90;
A = Table[Boole[Mod[n, k] == 0], {n, nn}, {k, nn}];
B = Table[If[Mod[k, n] == 0, MoebiusMu[n]*n, 0], {n, nn}, {k, nn}];
T = (A.B);
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 + 2]];
    F2 = Table[ If[Mod[n, k] == 0, b[[n/k]], 0], {n, nn}, {k, nn}];
    (F1.F2)[[All, 1]], {kk, nn - 2}];
TT = Transpose[TwinMangoldt];
Table[Sum[TT[[n, k]], {k, n}]/n, {n, nn - 2}]
(* This faster alternate conjectured program agrees with Antti Karttunen's precomputed list of numbers. *)
nn = 108; b = 4*Select[Range[1, nn, 2], SquareFreeQ]; bb = Table[DivisorSigma[0, n]*(MoebiusMu[n] + Sum[If[b[[j]] == n, LiouvilleLambda[n]*2/3, 0], {j, 1, Length[b]}]), {n, 1, nn}];
cc = Table[Sum[If[Mod[n, k] == 0, bb[[n/k]]*DivisorSigma[0, k], 0], {k, 1, n}], {n, 1, nn}] (* Mats Granvik, Mar 17 2019 *)
(* Dirichlet generating function *) s=7; nn=2500; N[Zeta[s]^2*Product[(1 - 2 Prime[j]^(-s)), {j, 1, nn}]*(1 + Sum[1/2/2^(n*(s - 1)), {n, 2, nn}]), 40] (* Mats Granvik, Apr 06 2019 *)

up_to = 256;
DirConv(ma,h) = { my(u = matsize(ma)[1], md = matrix(u,u)); for(n=1,u-h,for(k=1,u,md[n,k] = sumdiv(k,d,ma[n,d]*ma[n+h,k/d]))); (md); };
A298826list(up_to) = { my(h=2, matA = matrix(up_to+h,up_to+h,n,k,!(n%k)), matB = matrix(up_to+h,up_to+h,n,k,(!(k%n))*moebius(n)*n), matT = matA*matB, matD = DirConv(matT,2)); vector(up_to,i,(1/i)*sum(j=1,i,matD[j,i])); };
v298826 = A298826list(up_to);
A298826(n) = v298826[n]; \\ Antti Karttunen, Sep 30 2018

a(n) = Sum_{k=1..n} (A298674(k, n))/n = A298825(n)/n.
Conjecture: a(n) = (-1)^(n+1) * Sum_{d|n} A076479(d). - Daniel Suteu, Apr 04 2019
From Mats Granvik, Apr 06 2019: (Start)
The Dirichlet generating function, after Daniel Suteu above and Álvar Ibeas in A076479, appears to be: Sum_{n>=1} a(n)/n^s = zeta(s)^2*(Product_{j>=1} (1 - 2*prime(j)^(-s)))*(1 + Sum_{n>=2} ((1/2)/2^(n*(s - 1)))).
Conjectured formula: Let b(n) = 4*A056911(n) and c(n) = A000005(n)*A008683(n) + Sum_{j=1..length(b(1..N))} [b(j)=n]*A008836(n)*2/3) then a(n) = Sum_{k=1..n}[k|n] c(n/k)*A000005(k). (End)
Conjecture: a(n) = (-1)^(n+1) * Sum_{d|n} mu(d)*tau(d)*tau(n/d). - Ridouane Oudra, Nov 19 2019
The conjectured Dirichlet generating function simplifies to: Sum_{n>=1} a(n)/n^s = zeta(s)^2*(Product_{j>=1} (1 - 2*prime(j)^(-s)))*(1 + 2^(1 - s)/(2^s - 2)). - Steven Foster Clark, Sep 12 2022
Conjecture: abs(a(n)) = A361430(n). - Vaclav Kotesovec, Mar 12 2023
Conjecture: a(n) is multiplicative with a(p^e) = (-1)^p * (e-1) for prime p and e > 0. That conforms to the conjectured Dirichlet generating function (compare Steven Foster Clark, Sep 12 2022). - Werner Schulte, Jun 09 2025

More terms from Antti Karttunen, Sep 30 2018

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

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Jan 24 2018

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.

			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}
}

Terence Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. See second formula.

Cf. A191898, A298824, A001694, A298825, A298826.

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}]]

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.

A298824 Triangle read by rows. The transpose of the upper triangular part of the square matrix A298674.

Original entry on oeis.org

1, 2, -2, -1, 2, -1, 3, -1, 3, -1, 2, 2, -3, 2, -3, -2, -4, -2, 2, 4, 2, 2, 2, 2, 2, -5, 2, -5, 4, 0, 4, 0, 4, 0, 4, 0, -3, 3, -3, -3, 3, -3, -3, 3, -3, 4, -4, -6, -4, -6, -4, 4, 6, 4, 6, 2, 2, 2, 2, 2, 2, 2, 2, -9, 2, -9, -3, -2, -3, 1, 6, 1, -3, -2, -3, 1, 6, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, -11, 2, -11

Offset: 1

Mats Granvik, Jan 27 2018

			Triangle begins:
  ==========================================================
  n\k|  1   2   3   4   5   6   7   8   9  10   11  12   13
  ---+------------------------------------------------------
   1 |  1
   2 |  2  -2
   3 | -1   2  -1
   4 |  3  -1   3  -1
   5 |  2   2  -3   2  -3
   6 | -2  -4  -2   2   4   2
   7 |  2   2   2   2  -5   2  -5
   8 |  4   0   4   0   4   0   4   0
   9 | -3   3  -3  -3   3  -3  -3   3  -3
  10 |  4  -4  -6  -4  -6  -4   4   6   4   6
  11 |  2   2   2   2   2   2   2   2  -9   2   -9
  12 | -3  -2  -3   1   6   1  -3  -2  -3   1    6   1
  13 |  2   2   2   2   2   2   2   2   2   2  -11   2  -11

Row sums are A298825.

Cf. A298674.

Mathematica
```
Transpose[UpperTriangularize[A298674]]
```

T(n,k) = A298674(k,n), n <= k.

A306653 a(n) = Sum_{m=1..n} Sum_{k=1..n} [k divides n][n/k divides m]A008683(n/k)n/k[k divides m + 2^p]A008683(k)k, where p can be a positive integer: 1,2,3,4,5,...

Original entry on oeis.org

1, -2, -2, 2, -2, 4, -2, 0, 0, 4, -2, -4, -2, 4, 4, 0, -2, 0, -2, -4, 4, 4, -2, 0, 0, 4, 0, -4, -2, -8, -2, 0, 4, 4, 4, 0, -2, 4, 4, 0, -2, -8, -2, -4, 0, 4, -2, 0, 0, 0, 4, -4, -2, 0, 4, 0, 4, 4, -2, 8, -2, 4, 0, 0, 4, -8, -2, -4, 4, -8, -2, 0, -2, 4, 0, -4, 4, -8, -2, 0, 0, 4, -2, 8, 4

Offset: 1

Mats Granvik, Mar 03 2019

sign

It appears that for p=1 and p=2, a(n) is identically the same for all n except for n equal to multiples of 16. See A306652 for comparison.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Mats Granvik, Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture

Cf. A008683, A147848, A306652, A298825, A008683.

(* Dirichlet Convolution. *)
p=1;
a[n_] := 1/n*Sum[Sum[If[Mod[n, k] == 0, 1, 0]*If[Mod[m, n/k] == 0, 1, 0]*If[Mod[m + 2^p, k] == 0, 1, 0]*MoebiusMu[n/k]*n/k*MoebiusMu[k]*k, {k, 1, n}], {m, 1, n}]; a /@ Range[85]
(* conjectured faster program *)
nn = 85;
b = 4*Select[Range[1, nn, 2], SquareFreeQ];
bb = Table[DivisorSigma[0, n]*(MoebiusMu[n] + Sum[If[b[[j]] == n, LiouvilleLambda[n]*2/3, 0], {j, 1, Length[b]}]), {n, 1, nn}]

A306653(n) = (1/n)*sum(m=1, n, sumdiv(n, k, if( !(m%(n/k)) && !((m+(2^1))%k), n*moebius(n/k)*moebius(k),0))); \\ Antti Karttunen, Mar 15 2019

a(n) = 1/n * Sum_{m=1..n} Sum_{k=1..n} [k divides n]*[n/k divides m]*[k divides m + 2^p]*A008683(n/k)*n/k*A008683(k)*k, where p can be any positive integer: 1,2,3,4,5,...
a(n) = A000005(n)*(A008683(n) + Sum_{j=1..n} [4*A056911(j)=n]*A008836(n)*2/3) (conjectured formula verified for n=1..2500).
Dirichlet generating function, after Daniel Suteu in A298826 and Álvar Ibeas in A076479, appears to be: Sum_{n>=1} a(n)/n^s = (Product_{j>=1} (1 - 2*prime(j)^(-s)))*(1 + Sum_{n>=2} (1/2/2^(n*(s - 1)))). - Mats Granvik, Apr 07 2019
The conjectured Dirichlet generating function simplifies to: Sum_{n>=1} a(n)/n^s = (Product_{j>=1} (1 - 2*prime(j)^(-s)))*(1 + 2^(1 - s)/(2^s - 2)). - Steven Foster Clark, Sep 23 2022

A306652 a(n) = Sum_{m=1..n} Sum_{k=1..n} [k divides n][n/k divides m][k divides m + 4].

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 4, 10, 2, 4, 2, 8, 4, 4, 2, 12, 2, 4, 2, 8, 2, 8, 2, 14, 4, 4, 4, 8, 2, 4, 4, 12, 2, 8, 2, 8, 4, 4, 2, 20, 2, 4, 4, 8, 2, 4, 4, 12, 4, 4, 2, 16, 2, 4, 4, 14, 4, 8, 2, 8, 4, 8, 2, 12, 2, 4, 4, 8, 4, 8, 2, 20, 2, 4, 2, 16, 4

Offset: 1

Mats Granvik, Mar 03 2019

nonn

According to the first Hardy-Littlewood conjecture, the cousin primes have the same asymptotic density as the twin primes. In an analogy of Dirichlet convolutions A147848 corresponds to the twin primes while this sequence corresponds to the cousin primes, and it appears that this sequence differs from A147848 at multiples of 16. See A306653 for comparison.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Mats Granvik, Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture

Cf. A147848, A306652, A298825, A008683.

a[n_] := Sum[Sum[If[Mod[n, k] == 0, If[Mod[m, n/k] == 0, 1, 0], 0]*If[Mod[m + 4, k/1] == 0, 1, 0], {k, 1, n}], {m, 1, n}]; a /@Range[85] (* Dirichlet Convolution. *)
a[n_] := Sum[If[Mod[n, k] == 0, Sum[If[Mod[4, j] == 0, If[GCD[k, n/k] == j, j, 0], 0], {j, 1, n}], 0], {k, 1, n}]; a /@Range[85] (* GCD sum. *)
a[n_] := Sum[If[Mod[4, j] == 0, j*Count[Divisors[n], d_ /; GCD[d, n/d] == j], 0], {j, 1, n}]; a /@Range[85] (* After Jean-François Alcover in A034444. *)

A306652(n) = sum(m=1, n, sumdiv(n, k, !(m%(n/k)) && !((m+4)%k))); \\ Antti Karttunen, Mar 13 2019

a(n) = Sum_{m=1..n} Sum_{k=1..n} [k divides n]*[n/k divides m]*[k divides m + 4].

a(n) = Sum_{k=1..n}[k divides n]*Sum_{j=1..n}[j divides 4]*[GCD[k, n/k] = j]*j.

A316274 Nonzero terms in row sums of the lower triangular part of a square matrix formed by Dirichlet convolution of adjacent columns in the square matrix A191898.

Original entry on oeis.org

1, -4, -16, -9, -48, -25, -54, -128, 36, -49, -320, 144, -243, 100, 216, -121, -250, -768, 432, -169, 196, 400, 864, 225, -972, -1792, 1152, -289, 972, -686, -361, 784, 1200, 2592, 441, 484, 1000

Offset: 1

Mats Granvik, Jun 28 2018

sign

The motivation for this sequence is expression 1 in Terence Tao's blog post "Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges".

Cf. A191898.

Clear[nn, h, a, n, d, b, m];
nn = 500;
h = 1;
a[n_] := If[n < 1, 0, Sum[d MoebiusMu@d, {d, Divisors[n]}]];
TableForm[Transpose[Table[{n, a[n]}, {n, 1, nn}]]];
b = DeleteCases[
  Table[Sum[
    Sum[If[Mod[n, k] == 0, a[GCD[n/k, m]]*a[GCD[k, m + h]], 0], {k, 1,
       n}], {m, 1, n}], {n, 1, nn}], 0]

a(n) = A298825(A001694(n)). - Mats Granvik, Oct 08 2018

cp's OEIS Frontend

A298826 a(n) = A298825(n)/n.

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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).

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A298824 Triangle read by rows. The transpose of the upper triangular part of the square matrix A298674.

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A306653 a(n) = Sum_{m=1..n} Sum_{k=1..n} [k divides n]*[n/k divides m]*A008683(n/k)*n/k*[k divides m + 2^p]*A008683(k)*k, where p can be a positive integer: 1,2,3,4,5,...

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A306652 a(n) = Sum_{m=1..n} Sum_{k=1..n} [k divides n]*[n/k divides m]*[k divides m + 4].

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A316274 Nonzero terms in row sums of the lower triangular part of a square matrix formed by Dirichlet convolution of adjacent columns in the square matrix A191898.

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A306653 a(n) = Sum_{m=1..n} Sum_{k=1..n} [k divides n][n/k divides m]A008683(n/k)n/k[k divides m + 2^p]A008683(k)k, where p can be a positive integer: 1,2,3,4,5,...

A306652 a(n) = Sum_{m=1..n} Sum_{k=1..n} [k divides n][n/k divides m][k divides m + 4].