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 A191898 #160 Jan 28 2025 15:30:19
%S A191898 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,
%T A191898 -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,
%U A191898 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
%N A191898 Symmetric square array read by antidiagonals: T(n,1)=1, T(1,k)=1, T(n,k) = -Sum_{i=1..k-1} T(n-i,k) for n >= k, -Sum_{i=1..n-1} T(k-i,n) for n < k.
%C A191898 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 A023900 (conjecture). [The 2nd and 3rd formulas state that the conjecture is correct. _R. J. Mathar_, Sep 16 2017]
%C A191898 The sums from n=1 to infinity of T(n,k)/n converge to the Mangoldt function for column k (conjecture).
%C A191898 If gcd(n,k)=1 then T(n,k)=1 and if T(n,k)=1 then gcd(n,k)=1 (conjecture).
%C A191898 The Dirichlet generating functions for s > 1 for the columns appear to be (see A054535):
%C A191898 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))).
%C A191898 Examples:
%C A191898 k=1: Zeta(s)
%C A191898 k=2: Zeta(s)*(1 - 1/2^(s-1))
%C A191898 k=3: Zeta(s)*(1 - 1/3^(s-1))
%C A191898 k=4: Zeta(s)*(1 - 1/2^(s-1))
%C A191898 k=5: Zeta(s)*(1 - 1/5^(s-1))
%C A191898 k=6: Zeta(s)*(1 - 1/2^(s-1) - 1/3^(s-1) + 1/6^(s-1))
%C A191898 k=7: Zeta(s)*(1 - 1/7^(s-1))
%C A191898 ...
%C A191898 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))
%C A191898 ...
%C A191898 (conjecture)
%C A191898 See triangle A142971 for negative distinct prime factors.
%C A191898 This could probably be checked by matrix multiplication.
%C A191898 The signs of the eigenvalues of this matrix are a rearrangement of the Mobius function A008683 (conjecture). The first few eigenvalues are:
%C A191898 {1.0000}
%C A191898 {-1.4142, 1.4142}
%C A191898 {-2.6554, 1.8662, -1.2108}
%C A191898 {-3.4393, 2.1004, -1.6611, 0}
%C A191898 {-4.7711, -3.3867, 2.5910, -1.4332, 0}
%C A191898 {-5.2439, -3.4641, 3.4641, 2.5169, -2.2730, 0}
%C A191898 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]
%C A191898 From _Mats Granvik_, Jun 19 2016: (Start)
%C A191898 Remark about the Dirichlet generating function for the whole matrix: Subtracting the first column (in the form of zeta(c)) of the matrix gives us the limit: lim_{c->1} zeta(s)*zeta(c)/zeta(c+s-1)-zeta(c) = -zeta'(s)/zeta(s) which is the classical Dirichlet generating function for the von Mangoldt function.
%C A191898 For n >= k, see A231425, this matrix has row sums equal to zero except for the first row:
%C A191898 1=1
%C A191898 1-1=0
%C A191898 1+1-2=0
%C A191898 1-1+1-1=0
%C A191898 1+1+1+1-4=0
%C A191898 ...
%C A191898 log(A014963(n)) = Sum_{k>=1} A191898(n,k)/k, for n>1.
%C A191898 log(A014963(k)) = Sum_{n>=1} A191898(n,k)/n, for k>1.
%C A191898 log(A014963(n)) = limit of zeta(s)*(Sum_{d divides n} A008683(d)/d^(s-1)) as s->1, for n>1.
%C A191898 A008683(n) = Sum_{k=1..n} A191898(n,k)*exp(-i*2*Pi*k/n)/n.
%C A191898 A008683(n) = Sum_{n=1..k} A191898(n,k)*exp(-i*2*Pi*n/k)/k.
%C A191898 (End)
%C A191898 From _Mats Granvik_, Aug 09 2016: (Start)
%C A191898 The Dirichlet generating function for the matrix is zero for any pair c and s = 2 - c and any pair s and c = 2 - s except at the pole c = 1 and s = 1 where it is indeterminate.
%C A191898 In the Mathematica program section, in the expression for the matrix as Dirichlets characters, the variables s and c can apparently be any pair of positive integers.
%C A191898 Limits related to the Dirichlet generating function for the matrix: Let s = ZetaZero(n), then lim_{c->1} zeta(s*c)/zeta(c+s-1) = ZetaZero(n). Let s = ZetaZero(n), then lim_{c->1} zeta(s*c)/zeta(c+s*c-1) = ZetaZero(n)/(1+ZetaZero(n)).
%C A191898 (End)
%H A191898 Antti Karttunen, <a href="/A191898/b191898.txt">Table of n, a(n) for n = 1..22155; the first 210 antidiagonals of the array</a>
%H A191898 Mats Granvik, <a href="http://math.stackexchange.com/questions/424530/">Is this similarity to the Fourier transform of the von Mangoldt function real?</a>
%H A191898 Mats Granvik, <a href="http://mathoverflow.net/questions/162076/">Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?</a>
%H A191898 Mats Granvik, <a href="http://math.stackexchange.com/questions/156035/">Primes approximated by eigenvalues?</a>
%H A191898 Mats Granvik, <a href="http://math.stackexchange.com/questions/64194/">Are the primes found as a subset in this sequence a(n)?</a>
%H A191898 Mats Granvik, <a href="http://math.stackexchange.com/questions/1313321/">Will every eigenvalue in this type of matrix eventually be a common eigenvalue to infinitely many subsequent larger matrices of the same form?</a>
%H A191898 Mats Granvik, <a href="http://math.stackexchange.com/questions/228751/">How write Dirichlet character sums for the terms of the von Mangoldt function?</a>
%H A191898 Mats Granvik, <a href="http://math.stackexchange.com/questions/48946/">Do these series converge to the von Mangoldt function?</a>
%H A191898 Mats Granvik, <a href="http://math.stackexchange.com/questions/84177/">Is this sum equal to the Möbius function?</a>
%H A191898 Mats Granvik, <a href="http://math.stackexchange.com/questions/1133452/">Can the Riemann hypothesis be relaxed to say that this matrix A consists of square roots?</a>
%H A191898 Mats Granvik, <a href="http://math.stackexchange.com/questions/721418/elementary-proof-of-prime-number-theorem">Elementary proof of the prime number theorem?</a>
%H A191898 Mats Granvik, <a href="http://math.stackexchange.com/questions/1832268/is-this-dirichlet-series-generating-function-of-the-von-mangoldt-function-matrix">Is this Dirichlet series generating function of the von Mangoldt function matrix correct?</a>
%H A191898 Mats Granvik, <a href="https://math.stackexchange.com/q/4750704/8530">Question about ratios of polynomials evaluated at x=1</a>
%F A191898 T(n,1)=1, T(1,k)=1, n>=k: -Sum_{i=1..k-1} T(n-i,k), n<k: -Sum_{i=1..n-1} T(k-i,n).
%F A191898 T(n, n) = A023900(n). - _Michael Somos_, Jul 18 2011
%F A191898 T(n, k) = A023900(gcd(n,k)). - _Mats Granvik_, Jun 18 2012
%F A191898 Dirichlet generating function for sequence in the n-th row: zeta(s)*Sum_{ d divides n } mu(d)/d^(s-1). - _Mats Granvik_, Jun 18 2012 & Jun 19 2016
%F A191898 From _Mats Granvik_, Jun 19 2016: (Start)
%F A191898 Dirichlet generating function for the whole matrix: Sum_{k>=1} (Sum_{n>=1} T(n,k)/(n^c*k^s)) = Sum_{n>=1} (zeta(s)*Sum_{ d divides n } mu(d)/d^(s-1))/n^c = zeta(s)*zeta(c)/zeta( c + s - 1 ).
%F A191898 T(n,k) = A127093(n,k)^(1/2-i*a(k))*transpose(A008683(k)*(A127093(n,k)^(1/2+i*a(n)))) where a(x) is some real number. An example would be T(n,k) = A127093(n,k)^(zetazero(k))*transpose(A008683(k)*(A127093(n,k)^(zetazero(-k)))) but this is of course not special for only the zeta zeros.
%F A191898 Recurrence for a subset of A191898 that is a cross-directional variant of the recurrence in A051731: T(1,1)=1, T(1,2..k)=0, T(2..n,1)=0, n >= k: -Sum_{i=1..k-1} T(n-i,k) - T(n-i,k-1), n < k: -Sum_{i=1..n-1} T(k-i,n) - T(k-i,n-1). Notice that the identity matrix in linear algebra satisfies a similar recurrence:
%F A191898 T(1,1)=1, T(1,2..k)=0, T(2..n,1)=0, n >= k: -Sum_{i=1..n-1} T(n-i,k) - T(n-i,k-1), n < k: -Sum_{i=1..k-1} T(k-i,n) - T(k-i,n-1).
%F A191898 (End)
%F A191898 This array equals A051731*transpose(A143256). - _Mats Granvik_, Jul 22 2016
%F A191898 T(n,k) = sqrt(A143256(n,k))*transpose(sqrt(A143256(n,k))). - _Mats Granvik_, Aug 10 2018
%F A191898 Dirichlet generating function for absolute values: Sum_{k>=1} (Sum_{n>=1} abs(T(n,k))/(n^c*k^s)) = zeta(s)*zeta(c)*zeta(s + c - 1)/zeta(2*(s + c - 1))*Product_{k>=1} (1 - 2/(prime(k) + prime(k)^(s + c))). After Vaclav Kotesovec in A173557. - _Mats Granvik_, Apr 25 2021
%e A191898 Array starts:
%e A191898 n\k | 1 2 3 4 5 6 7 8 9 10
%e A191898 ----+-----------------------------------------------------
%e A191898 1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A191898 2 | 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ...
%e A191898 3 | 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, ...
%e A191898 4 | 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ...
%e A191898 5 | 1, 1, 1, 1, -4, 1, 1, 1, 1, -4, ...
%e A191898 6 | 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, ...
%e A191898 7 | 1, 1, 1, 1, 1, 1, -6, 1, 1, 1, ...
%e A191898 8 | 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ...
%e A191898 9 | 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, ...
%e A191898 10 | 1, -1, 1, -1, -4, -1, 1, -1, 1, 4, ...
%t A191898 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 *)
%t A191898 (* Conjectured expression for the matrix as Dirichlet characters *) s = RandomInteger[{1, 3}]; c = RandomInteger[{1, 3}]; nn = 12; b = Table[Exp[MangoldtLambda[Divisors[n]]]^-MoebiusMu[Divisors[n]], {n, 1, nn^Max[s, c]}]; j = 1; MatrixForm[Table[Table[Product[(b[[n^s]][[m]]*DirichletCharacter[b[[n^s]][[m]], j, k^c] - (b[[n^s]][[m]] - 1)), {m, 1, Length[Divisors[n]]}], {n, 1, nn}], {k, 1, nn}]] (* _Mats Granvik_, Nov 23 2013 and Aug 09 2016 *)
%o A191898 (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 */
%o A191898 (Python)
%o A191898 from sympy.core.cache import cacheit
%o A191898 @cacheit
%o A191898 def T(n, k): return 0 if n<1 or k<1 else 1 if n==1 or k==1 else T(k, n) if k>n else T(k, (n - 1)%k + 1) if n>k else -sum([T(n, i) for i in range(1, n)])
%o A191898 for n in range(1, 21): print([T(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Oct 23 2017
%Y A191898 Cf. A023900, A014963, A008302, A033999, A061347, A142971.
%Y A191898 Cf. A007917, A199514, A260237. - _Mats Granvik_, Jun 19 2016
%Y A191898 Cf. A343555, A343556.
%K A191898 sign,tabl
%O A191898 1,13
%A A191898 _Mats Granvik_, Jun 19 2011