cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A298825 Row sums of A298824.

Original entry on oeis.org

1, 0, 0, 4, 0, 0, 0, 16, -9, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, -25, 0, -54, 0, 0, 0, 0, 128, 0, 0, 0, -36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -49, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 320, 0, 0, 0, 0, 0, 0, 0, -144, 0, 0, 0, 0, 0, 0, 0, 0, -243, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Mats Granvik, Jan 27 2018

Keywords

Comments

Positions of nonzero entries appear to be given by A001694.

Links

Crossrefs

Cf. A298674, A298824, A298826.

Programs

  • Mathematica
    a[n_] := n*Sum[If[Mod[n, j] == 0,DivisorSigma[0, n/j]*1/j*Sum[Sum[If[Mod[j, k] == 0, If[Mod[m, j/k] == 0, MoebiusMu[j/k]*j/k, 0], 0]*If[Mod[m + 2, k/1] == 0, MoebiusMu[k/1]*k/1, 0], {k, 1, j}], {m, 1, j}], 0], {j, 1, n}]; a /@ Range[85] (* Mats Granvik, Mar 03 2019 *)
  • PARI
    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); };
A298825list(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,sum(j=1,i,matD[j,i])); };
v298825 = A298825list(up_to);
A298825(n) = v298825[n]; \\ Antti Karttunen, Sep 30 2018

Formula

From Mats Granvik, Mar 03 2019: (Start)
a(n) = n*Sum_{j=1..n} [j divides n]*A000005(n/j)*A306653(j).
a(n) = n*Sum_{j=1..n} [j divides n]*A000005(n/j)*Sum_{m=1..n} Sum_{k=1..n} [k divides j]*[j/k divides m]*A008683(j/k)*j/k*[k divides m + 2^p]*A008683(k)*k, p=1,2,3,4,5,...,infinity.
(End)

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

Views

Author

Mats Granvik, Mar 03 2019

Keywords

Comments

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.

Links

Crossrefs

Cf. A147848, A306652, A298825, A008683.

Programs

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

Formula

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.
Showing 1-2 of 2 results.

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