A275205 -id:A275205 - cp's OEIS frontend

A309229 Square array read by upwards antidiagonals: T(n,k) = Sum_{i=1..n} A191898(i,k).

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, 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, 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

Offset: 1

Mats Granvik, Aug 10 2019

log(A003418(n)) = Sum_{k>=1} (T(n, k)/k - 1/k).
Partial sums of the symmetric matrix A191898. - Mats Granvik, Apr 12 2020
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
From Mats Granvik, Apr 14 2021: (Start)
Conjecture 1: For n>1: max(T(1..n,n)) + min(T(1..n,n)) = 2*mean(T(1..n,n)) = -A023900(n).
Patterns that eventually fail or possibly become switched are:
max(T(n,1..n!)) = 1,2,3,4,5,6,7,8,...
min(T(n,1..n!)) = 1,0,-2,-3,-7,-5,-11,-12,...
which are the first 8 terms of A275205.
Conjecture 2: The Prime Number Theorem should imply: mean(T(n,1..n!)) = 1.
(End)

			   1, 1, 1, 1, 1,  1, 1, 1, 1,  1,  1,  1,  1,  1, ...
   2, 0, 2, 0, 2,  0, 2, 0, 2,  0,  2,  0,  2,  0, ...
   3, 1, 0, 1, 3, -2, 3, 1, 0,  1,  3, -2,  3,  1, ...
   4, 0, 1, 0, 4, -3, 4, 0, 1,  0,  4, -3,  4,  0, ...
   5, 1, 2, 1, 0, -2, 5, 1, 2, -4,  5, -2,  5,  1, ...
   6, 0, 0, 0, 1,  0, 6, 0, 0, -5,  6,  0,  6,  0, ...
   7, 1, 1, 1, 2,  1, 0, 1, 1, -4,  7,  1,  7, -6, ...
   8, 0, 2, 0, 3,  0, 1, 0, 2, -5,  8,  0,  8, -7, ...
   9, 1, 0, 1, 4, -2, 2, 1, 0, -4,  9, -2,  9, -6, ...
  10, 0, 1, 0, 0, -3, 3, 0, 1,  0, 10, -3, 10, -7, ...
  11, 1, 2, 1, 1, -2, 4, 1, 2,  1,  0, -2, 11, -6, ...
  12, 0, 0, 0, 2,  0, 5, 0, 0,  0,  1,  0, 12, -7, ...
  13, 1, 1, 1, 3,  1, 6, 1, 1,  1,  2,  1,  0, -6, ...
  14, 0, 2, 0, 4,  0, 0, 0, 2,  0,  3,  0,  1,  0, ...
  ...

Mats Granvik, Attempt at proof of the conjectured square root order asymptotics for the sequence constructed from this matrix.
Mats Granvik, Mathematica MatrixPlot of 1000 times 1000 size matrix
Mats Granvik, Mathematica program for the recurrence
Mats Granvik, Mathematica program to compute the sequence with the conjectured asymptotic sqrt(8*n)
Mathematics Stack Exchange, Do these series converge to the von Mangoldt function?

Cf. A003418, A191898.

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 *)

Recurrence:
T(n, 1) = [n >= 1]*n;
T(1, k) = 1;
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
T(n,k) = Sum_{i=1..n} A191898(i,k).

A309192 a(n) = Sum_{k=1..n} mu(k)^2 * k * floor(n/k).

Original entry on oeis.org

1, 4, 8, 11, 17, 29, 37, 40, 44, 62, 74, 86, 100, 124, 148, 151, 169, 181, 201, 219, 251, 287, 311, 323, 329, 371, 375, 399, 429, 501, 533, 536, 584, 638, 686, 698, 736, 796, 852, 870, 912, 1008, 1052, 1088, 1112, 1184, 1232, 1244, 1252, 1270, 1342, 1384, 1438, 1450, 1522

Offset: 1

Ilya Gutkovskiy, Jul 16 2019

nonn

Partial sums of A048250.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A008683, A024916, A024924, A048250, A064608, A275205.

Table[Sum[MoebiusMu[k]^2 k Floor[n/k], {k, 1, n}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[1/(1 - x) Sum[MoebiusMu[k]^2 k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Accumulate[Table[Total[Select[Divisors[n], SquareFreeQ]], {n, 1, 100}]] (* Vaclav Kotesovec, Jul 16 2019 *)

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k)^2 * k * x^k/(1 - x^k).

a(n) ~ n^2/2. - Vaclav Kotesovec, Jul 16 2019

cp's OEIS Frontend

A309229 Square array read by upwards antidiagonals: T(n,k) = Sum_{i=1..n} A191898(i,k).

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