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

Author

Mats Granvik, Aug 10 2019

Keywords

• tabl
• sign

Comments

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)

Examples

			   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, ...
  ...
		

Links

• 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?

Crossrefs

Cf. A003418, A191898.

Programs

Mathematica

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 *)

Formula

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).