This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309229 #110 May 24 2021 01:03:24
%S A309229 1,2,1,3,0,1,4,1,2,1,5,0,0,0,1,6,1,1,1,2,1,7,0,2,0,3,0,1,8,1,0,1,4,-2,
%T A309229 2,1,9,0,1,0,0,-3,3,0,1,10,1,2,1,1,-2,4,1,2,1,11,0,0,0,2,0,5,0,0,0,1,
%U A309229 12,1,1,1,3,1,6,1,1,1,2,1,13,0,2,0,4,0,0,0,2,0,3,0,1,14,1,0,1,0,-2,1,1,0,-4,4,-2,2,1
%N A309229 Square array read by upwards antidiagonals: T(n,k) = Sum_{i=1..n} A191898(i,k).
%C A309229 log(A003418(n)) = Sum_{k>=1} (T(n, k)/k - 1/k).
%C A309229 Partial sums of the symmetric matrix A191898. - _Mats Granvik_, Apr 12 2020
%C A309229 1 + Sum_{k=1..2*n} sign((sign(n+Sum_{j=2..k}-|T(n,j)|)+1)) appears to be asymptotic to sqrt(8*n). - _Mats Granvik_, Jun 08 2020
%C A309229 From _Mats Granvik_, Apr 14 2021: (Start)
%C A309229 Conjecture 1: For n>1: max(T(1..n,n)) + min(T(1..n,n)) = 2*mean(T(1..n,n)) = -A023900(n).
%C A309229 Patterns that eventually fail or possibly become switched are:
%C A309229 max(T(n,1..n!)) = 1,2,3,4,5,6,7,8,...
%C A309229 min(T(n,1..n!)) = 1,0,-2,-3,-7,-5,-11,-12,...
%C A309229 which are the first 8 terms of A275205.
%C A309229 Conjecture 2: The Prime Number Theorem should imply: mean(T(n,1..n!)) = 1.
%C A309229 (End)
%H A309229 Mats Granvik, <a href="https://mathoverflow.net/a/387056/25104">Attempt at proof of the conjectured square root order asymptotics for the sequence constructed from this matrix.</a>
%H A309229 Mats Granvik, <a href="/A309229/a309229.png">Mathematica MatrixPlot of 1000 times 1000 size matrix</a>
%H A309229 Mats Granvik, <a href="/A309229/a309229_2.txt">Mathematica program for the recurrence</a>
%H A309229 Mats Granvik, <a href="/A309229/a309229_1.txt">Mathematica program to compute the sequence with the conjectured asymptotic sqrt(8*n)</a>
%H A309229 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/48946/8530">Do these series converge to the von Mangoldt function?</a>
%F A309229 Recurrence:
%F A309229 T(n, 1) = [n >= 1]*n;
%F A309229 T(1, k) = 1;
%F A309229 T(n, k) = [n > k]*T(n - k, k) + [n <= k](Sum_{i=0..n-1} T(n - 1, k - i) - Sum_{i=1..n-1} T(n, k - i)). - _Mats Granvik_, Jun 19 2020
%F A309229 T(n,k) = Sum_{i=1..n} A191898(i,k).
%e A309229 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A309229 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...
%e A309229 3, 1, 0, 1, 3, -2, 3, 1, 0, 1, 3, -2, 3, 1, ...
%e A309229 4, 0, 1, 0, 4, -3, 4, 0, 1, 0, 4, -3, 4, 0, ...
%e A309229 5, 1, 2, 1, 0, -2, 5, 1, 2, -4, 5, -2, 5, 1, ...
%e A309229 6, 0, 0, 0, 1, 0, 6, 0, 0, -5, 6, 0, 6, 0, ...
%e A309229 7, 1, 1, 1, 2, 1, 0, 1, 1, -4, 7, 1, 7, -6, ...
%e A309229 8, 0, 2, 0, 3, 0, 1, 0, 2, -5, 8, 0, 8, -7, ...
%e A309229 9, 1, 0, 1, 4, -2, 2, 1, 0, -4, 9, -2, 9, -6, ...
%e A309229 10, 0, 1, 0, 0, -3, 3, 0, 1, 0, 10, -3, 10, -7, ...
%e A309229 11, 1, 2, 1, 1, -2, 4, 1, 2, 1, 0, -2, 11, -6, ...
%e A309229 12, 0, 0, 0, 2, 0, 5, 0, 0, 0, 1, 0, 12, -7, ...
%e A309229 13, 1, 1, 1, 3, 1, 6, 1, 1, 1, 2, 1, 0, -6, ...
%e A309229 14, 0, 2, 0, 4, 0, 0, 0, 2, 0, 3, 0, 1, 0, ...
%e A309229 ...
%t A309229 f[n_] := DivisorSum[n, MoebiusMu[#] # &]; nn = 14; A = Accumulate[Table[Table[f[GCD[n, k]], {k, 1, nn}], {n, 1, nn}]]; Flatten[Table[Table[A[[n - k + 1, k]], {k, 1, n}], {n, 1, nn}]] (* _Mats Granvik_, Jun 09 2020 *)
%Y A309229 Cf. A003418, A191898.
%K A309229 tabl,sign
%O A309229 1,2
%A A309229 _Mats Granvik_, Aug 10 2019