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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).

%I A253951 #35 Mar 20 2015 21:15:26
%S A253951 0,1,5,9,20,23,42,52,69,77,113,119,165,177,190,214,279,291,366,379,
%T A253951 399,422,517,533,599,625,679,701,829,846,986,1035,1069,1105,1137,1164,
%U A253951 1339,1380,1417,1449,1646,1674,1883,1918,1955,2008,2239,2274,2420,2462,2515,2559,2827,2874,2929
%N 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).
%C A253951 a(n) ~ log(A003418(n))*n, based on the comment by _Hans Havermann_ in A048272 referring to an argument by Gareth McCaughan.
%C A253951 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.
%C A253951 Compare a(n) to round(log(A003418)*n)= 0, 1, 5, 10, 20, 25, 42, 54, 70, 78,...
%H A253951 Robert G. Wilson v, <a href="/A253951/b253951.txt">Table of n, a(n) for n = 1..1002</a>
%F A253951 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)
%p A253951 with(LinearAlgebra):
%p A253951 N:= 200:
%p A253951 A051731:= Matrix(N,N,(n,k) -> `if`(n mod k = 0, 1, 0),shape=triangular[lower]):
%p A253951 A127093:= Matrix(N,N,(n,k) -> `if`(n mod k = 0, k, 0), shape=triangular[lower]):
%p A253951 A054524T:= Matrix(N,N,(k,n) -> `if`(n mod k = 0, numtheory:-mobius(k),0), shape=triangular[upper]):
%p A253951 T:= A051731 . A127093 . A054524T:
%p A253951 a[1]:= 0:
%p A253951 for n from 2 to N do
%p A253951   a[n]:= a[n-1] + add(T[i,n],i=1..n) + add(T[n,j],j=2..n-1)
%p A253951 od:
%p A253951 seq(a[n],n=1..N); # _Robert Israel_, Jan 20 2015
%t A253951 nn = 55;
%t A253951 Z = Table[ If[ Mod[n, k] == 0, 1, 0], {n, nn}, {k, nn}];
%t A253951 A = Table[ If[ Mod[n, k] == 0, k, 0], {n, nn}, {k, nn}];
%t A253951 B = Table[ If[ Mod[n, k] == 0, MoebiusMu[k], 0], {n, nn}, {k, nn}];
%t A253951 MatrixForm[T = Z.A.Transpose[B]];
%t A253951 T[[All, 1]] = 0;
%t A253951 a = Table[ Total[ T[[1 ;; n, 1 ;; n]], 2], {n, nn}]
%t A253951 (* shows a graph *) Show[ ListLinePlot[a], ListLinePlot[ Accumulate[ MangoldtLambda[ Range[ nn]]]]]
%K A253951 nonn
%O A253951 1,3
%A A253951 _Mats Granvik_, Jan 20 2015