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

1,3

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

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)

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]]]]]

Crossrefs

Sequence in context: A292773 A228338 A309731 * A102172 A011983 A087940

Adjacent sequences: A253948 A253949 A253950 * A253952 A253953 A253954

Keyword

nonn

Author

Mats Granvik, Jan 20 2015

Status

approved