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A191898 Symmetric square array read by anti-diagonals. T(n,1)=1, T(1,k)=1, n>=k: -Sum from i=1 to k-1 of T(n-i,k), n<k: -Sum from i=1 to n-1 of T(k-i,n). 17
1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -2, 1, 1, -2, 1, 1, 1, -1, 1, -1, -4, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -6, -1, 1, -1, 1, -1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,13

COMMENTS

Rows equal columns and are periodic. No zero elements are found (conjecture). The recurrence is related to the recurrence for the Mahonian numbers. The main diagonal is the Dirichlet inverse of the Euler totient function (conjecture).

The sums from n=1 to infinity of T(n,k)/n converge to the Mangoldt function for column k (conjecture).

If gcd(n,k)=1 then T(n,k)=1 and if T(n,k)=1 then gcd(n,k)=1 (conjecture).

The Dirichlet generating functions for s>1 for the columns appear to be (see A054535):

Zeta(s)*(1 + (Sum over all possible combinations of products of negative distinct prime factors of k, up to rearrangement, 1/((-1* first distinct prime factor)*(-1*second distinct prime factor)*(-1*third distinct prime factor * ...))^(s-1))).

Examples:

k=1: Zeta(s)

k=2: Zeta(s)*(1-1/2^(s-1))

k=3: Zeta(s)*(1-1/3^(s-1))

k=4: Zeta(s)*(1-1/2^(s-1))

k=5: Zeta(s)*(1-1/5^(s-1))

k=6: Zeta(s)*(1-1/2^(s-1)-1/3^(s-1)+1/6^(s-1))

k=7: Zeta(s)*(1-1/7^(s-1))

...

k=30: Zeta(s)*(1-1/2^(s-1)-1/3^(s-1)-1/5^(s-1)+1/6^(s-1)+1/10^(s-1)+1/15^(s-1)-1/30^(s-1))

...

(conjecture)

See triangle A142971 for negative distinct prime factors.

This could probably be checked by matrix multiplication.

The signs of the eigenvalues of this matrix are a rearrangement of the Mobius function A008683 (conjecture). The first few eigenvalues are:

{1.0000}

{-1.4142, 1.4142}

{-2.6554, 1.8662, -1.2108}

{-3.4393, 2.1004, -1.6611, 0}

{-4.7711, -3.3867, 2.5910, -1.4332, 0}

{-5.2439, -3.4641, 3.4641, 2.5169, -2.2730, 0}

The relation to Dirichlet characters for the entries in this matrix appears appears to be formulated in terms of the sequence A089026 which is equal to n if n is a prime, otherwise equal to 1. See Mathematica program below. [Mats Granvik, Nov 23 2013]

LINKS

Table of n, a(n) for n=1..91.

Mats Granvik, Is this similarity to the Fourier transform of the von Mangoldt function real?

Mats Granvik, Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mats Granvik, Primes approximated by eigenvalues?

Mats Granvik, Are the primes found as a subset in this sequence a(n)?

Mats Granvik, Will every eigenvalue in this type of matrix eventually be a common eigenvalue to infinitely many subsequent larger matrices of the same form?

Mats Granvik, How write Dirichlet character sums for the terms of the von Mangoldt function?

Mats Granvik, Do these series converge to the von Mangoldt function?

Mats Granvik, Is this sum equal to the Möbius function?

Mats Granvik, Can the Riemann hypothesis be relaxed to say that this matrix A consists of square roots?

FORMULA

T(n,1)=1, T(1,k)=1, n>=k: -Sum from i=1 to k-1 of T(n-i,k), n<k: -Sum from i=1 to n-1 of T(k-i,n).

T(n, n) = A023900(n). - Michael Somos, Jul 18 2011

T(n,k) = A023900(GCD(n,k)). Conjecture: The Dirichlet generating function for the n-th row is Zeta[s] Total[1/Divisors[n]^(s - 1) * MoebiusMu[Divisors[n]]]]. - Mats Granvik, Jun 18 2012

EXAMPLE

Array starts:

1..1..1..1..1..1..1..1..1..1...

1.-1..1.-1..1.-1..1.-1..1.-1...

1..1.-2..1..1.-2..1..1.-2..1...

1.-1..1.-1..1.-1..1.-1..1.-1...

1..1..1..1.-4..1..1..1..1.-4...

1.-1.-2.-1..1..2..1.-1.-2.-1...

1..1..1..1..1..1.-6..1..1..1...

1.-1..1.-1..1.-1..1.-1..1.-1...

1..1.-2..1..1.-2..1..1.-2..1...

1.-1..1.-1.-4.-1..1.-1..1..4...

MATHEMATICA

T[ n_, k_] := T[ n, k] = Which[ n < 1 || k < 1, 0, n == 1 || k == 1, 1, k > n, T[k, n], n > k, T[k, Mod[n, k, 1]], True, -Sum[ T[n, i], {i, n - 1}]]; (* Michael Somos, Jul 18 2011 *)

nn = 12; b = Table[Exp[MangoldtLambda[Divisors[n]]]^-MoebiusMu[Divisors[n]], {n, 1, nn}]; j = 1; MatrixForm[Table[Table[Product[(b[[n]][[m]] * DirichletCharacter[b[[n]][[m]], j, k] - (b[[n]][[m]] - 1)), {m, 1, Length[Divisors[n]]}], {n, 1, nn}], {k, 1, nn}]] (* Conjectured expression as Dirichlet characters. Mats Granvik, Nov 23 2013 *)

PROG

(PARI) {T(n, k) = if( n<1 || k<1, 0, n==1 || k==1, 1, k>n, T(k, n), k<n, T(k, (n-1)%k+1), -sum( i=1, n-1, T(n, i)))}; /* Michael Somos, Jul 18 2011 */

CROSSREFS

Cf. A023900, A014963, A008302, A033999, A061347, A142971.

Sequence in context: A113515 A103754 A058665 * A043290 A194333 A203640

Adjacent sequences:  A191895 A191896 A191897 * A191899 A191900 A191901

KEYWORD

sign,tabl

AUTHOR

Mats Granvik, Jun 19 2011

STATUS

approved

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Last modified June 30 12:56 EDT 2015. Contains 259077 sequences.