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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). 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 5 16:53 EDT 2020. Contains 333245 sequences. (Running on oeis4.)