A298826 - OEIS

%I #119 Mar 13 2023 06:13:41

%S 1,0,0,1,0,0,0,2,-1,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,-1,0,-2,0,0,0,0,4,0,

%T 0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,

%U 0,0,0,0,0,-2,0,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2

%N a(n) = A298825(n)/n.

%C From _Mats Granvik_, Apr 06 2019: (Start)

%C Positions of nonzero entries appear to be given by A001694.

%C The limit lim_{k->oo} Sum_{n=1..k} a(n)/n appears to converge to some number.

%C Sum_{n=1..30000} a(n)/n = 1.31897... This appears to be in agreement with:

%C (1/30000)*Sum_{n<=30000} log(A014963(n))*log(A014963(n+2)) = 1.30351...

%C If the limit can be proven to converge to a number greater than 1, then it is true that Sum_{n<=X} log(A014963(n))*log(A014963(n+2)) > X, where ">" means "greater than" as in usual inequality notation.

%C The twin prime conjecture, according to Terence Tao on his blog, is that Sum_{n<=X} log(A014963(n))*log(A014963(n+2)) >> X where ">>" means "asymptotically greater than". There he also says that the first Hardy Littlewood conjecture states that Sum_{n<=X} log(A014963(n))*log(A014963(n+h)) = S(h)*X + o(X), where "S(h)" is the singular series.

%C Compare this to the prime number theorem which is lim_{X->oo} (Sum_{n<=X} log(A014963(n)))/X = 1.

%C (End)

%H Antti Karttunen, <a href="/A298826/b298826.txt">Table of n, a(n) for n = 1..2500</a>

%H Mats Granvik, <a href="https://mathoverflow.net/q/308990/25104">Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture</a>.

%H Terence Tao, <a href="https://terrytao.wordpress.com/2017/07/06/correlations-of-the-von-mangoldt-and-higher-divisor-functions-i-long-shift-ranges/">Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges</a>. See second formula.

%F a(n) = Sum_{k=1..n} (A298674(k, n))/n = A298825(n)/n.

%F Conjecture: a(n) = (-1)^(n+1) * Sum_{d|n} A076479(d). - _Daniel Suteu_, Apr 04 2019

%F From _Mats Granvik_, Apr 06 2019: (Start)

%F The Dirichlet generating function, after Daniel Suteu above and Álvar Ibeas in A076479, appears to be: Sum_{n>=1} a(n)/n^s = zeta(s)^2*(Product_{j>=1} (1 - 2*prime(j)^(-s)))*(1 + Sum_{n>=2} ((1/2)/2^(n*(s - 1)))).

%F Conjectured formula: Let b(n) = 4*A056911(n) and c(n) = A000005(n)*A008683(n) + Sum_{j=1..length(b(1..N))} [b(j)=n]*A008836(n)*2/3) then a(n) = Sum_{k=1..n}[k|n] c(n/k)*A000005(k). (End)

%F Conjecture: a(n) = (-1)^(n+1) * Sum_{d|n} mu(d)*tau(d)*tau(n/d). - _Ridouane Oudra_, Nov 19 2019

%F The conjectured Dirichlet generating function simplifies to: Sum_{n>=1} a(n)/n^s = zeta(s)^2*(Product_{j>=1} (1 - 2*prime(j)^(-s)))*(1 + 2^(1 - s)/(2^s - 2)). - _Steven Foster Clark_, Sep 12 2022

%F Conjecture: abs(a(n)) = A361430(n). - _Vaclav Kotesovec_, Mar 12 2023

%t nn = 90;

%t A = Table[Boole[Mod[n, k] == 0], {n, nn}, {k, nn}];

%t B = Table[If[Mod[k, n] == 0, MoebiusMu[n]*n, 0], {n, nn}, {k, nn}];

%t T = (A.B);

%t TwinMangoldt = Table[a = T[[All, kk]];

%t     F1 = Table[If[Mod[n, k] == 0, a[[n/k]], 0], {n, nn}, {k, nn}];

%t     b = T[[All, kk + 2]];

%t     F2 = Table[ If[Mod[n, k] == 0, b[[n/k]], 0], {n, nn}, {k, nn}];

%t     (F1.F2)[[All, 1]], {kk, nn - 2}];

%t TT = Transpose[TwinMangoldt];

%t Table[Sum[TT[[n, k]], {k, n}]/n, {n, nn - 2}]

%t (* From _Mats Granvik_, Mar 17 2019: (Start) *)

%t nn = 108; b = 4*Select[Range[1, nn, 2], SquareFreeQ]; bb = Table[DivisorSigma[0, n]*(MoebiusMu[n] + Sum[If[b[[j]] == n, LiouvilleLambda[n]*2/3, 0], {j, 1, Length[b]}]), {n, 1, nn}];

%t cc = Table[Sum[If[Mod[n, k] == 0, bb[[n/k]]*DivisorSigma[0, k], 0], {k, 1,

%t     n}], {n, 1, nn}]

%t (* This faster conjectured program agrees with Antti Karttunen's precomputed \

%t list of numbers. *)(* (End) *)

%t (* Dirichlet generating function *) s=7; nn=2500; N[Zeta[s]^2*Product[(1 - 2 Prime[j]^(-s)), {j, 1, nn}]*(1 + Sum[1/2/2^(n*(s - 1)), {n, 2, nn}]), 40] (* _Mats Granvik_, Apr 06 2019 *)

%o (PARI)

%o up_to = 256;

%o 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); };

%o A298826list(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,(1/i)*sum(j=1,i,matD[j,i])); };

%o v298826 = A298826list(up_to);

%o A298826(n) = v298826[n]; \\ _Antti Karttunen_, Sep 30 2018

%Y Cf. A298674, A298824, A298825, A001694, A361430.

%K sign

%O 1,8

%A _Mats Granvik_, Jan 27 2018

%E More terms from _Antti Karttunen_, Sep 30 2018