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

1,2

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)

Links

Table of n, a(n) for n=1..105.

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?

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).

Example

   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, ...
  ...

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 *)

Crossrefs

Cf. A003418, A191898.

Sequence in context: A127094 A221642 A158906 * A143239 A158951 A126988

Adjacent sequences: A309226 A309227 A309228 * A309230 A309231 A309232

Keyword

tabl,sign

Author

Mats Granvik, Aug 10 2019

Status

approved