A253951 A partial double sum of integers: a(n) = Sum_{x=1..n} Sum_{y=1..n} T(x,y), where T is the matrix product: T = A051731*A127093*Transpose(A054524) and T(n,1)=0 (* stands for matrix multiplication).

0, 1, 5, 9, 20, 23, 42, 52, 69, 77, 113, 119, 165, 177, 190, 214, 279, 291, 366, 379, 399, 422, 517, 533, 599, 625, 679, 701, 829, 846, 986, 1035, 1069, 1105, 1137, 1164, 1339, 1380, 1417, 1449, 1646, 1674, 1883, 1918, 1955, 2008, 2239, 2274, 2420, 2462, 2515, 2559, 2827, 2874, 2929

Offset: 1

Author

Mats Granvik, Jan 20 2015

Keywords

• nonn

Comments

a(n) ~ log(A003418(n))*n, based on the comment by Hans Havermann in A048272 referring to an argument by Gareth McCaughan.

The exact relation is: lim_{n->Infinity} log(A003418(k))*n = Sum_{x=1..n} Sum_{y=1..k} T(x,y), where T is the matrix product: T = A051731*A127093*Transpose(A054524) and T(n,1)=0.

Compare a(n) to round(log(A003418)*n)= 0, 1, 5, 10, 20, 25, 42, 54, 70, 78,...

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1002

Programs

Maple

with(LinearAlgebra):
N:= 200:
A051731:= Matrix(N,N,(n,k) -> `if`(n mod k = 0, 1, 0),shape=triangular[lower]):
A127093:= Matrix(N,N,(n,k) -> `if`(n mod k = 0, k, 0), shape=triangular[lower]):
A054524T:= Matrix(N,N,(k,n) -> `if`(n mod k = 0, numtheory:-mobius(k),0), shape=triangular[upper]):
T:= A051731 . A127093 . A054524T:
a[1]:= 0:
for n from 2 to N do
  a[n]:= a[n-1] + add(T[i,n],i=1..n) + add(T[n,j],j=2..n-1)
od:
seq(a[n],n=1..N); # Robert Israel, Jan 20 2015

Mathematica

nn = 55;
Z = Table[ If[ Mod[n, k] == 0, 1, 0], {n, nn}, {k, nn}];
A = Table[ If[ Mod[n, k] == 0, k, 0], {n, nn}, {k, nn}];
B = Table[ If[ Mod[n, k] == 0, MoebiusMu[k], 0], {n, nn}, {k, nn}];
MatrixForm[T = Z.A.Transpose[B]];
T[[All, 1]] = 0;
a = Table[ Total[ T[[1 ;; n, 1 ;; n]], 2], {n, nn}]
(* shows a graph *) Show[ ListLinePlot[a], ListLinePlot[ Accumulate[ MangoldtLambda[ Range[ nn]]]]]

Formula

a(n) = Sum_{x=1..n} Sum_{y=1..n} T(x,y), where T is the matrix product: T=A051731*A127093*Transpose(A054524) and T(n,1)=0. (* stands for matrix multiplication)