A204120 Symmetric matrix based on f(i,j) = gcd(prime(i+1),prime(j+1)), by antidiagonals.

3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Clark Kimberling, Jan 11 2012

Keywords

• nonn
• tabl

Comments

A204120 represents the matrix M given by f(i,j)=GCD(prime(i+1),prime(j+1)) for i>=1 and j>=1. See A204121 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

Square array with odd primes (A065091) on main diagonal, and 1 at all other entries; array A204118 without its top row and the leftmost column. - Antti Karttunen, Sep 25 2018

Examples

			Northwest corner:
3 1 1 1
1 5 1 1
1 1 7 1
1 1 1 11
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65703 (the first 362 antidiagonals of array)

Crossrefs

Cf. A065091 (main diagonal), A204118, A204121, A204016, A202453.

Programs

Mathematica

f[i_, j_] := GCD[Prime[i + 1], Prime[j + 1]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]    (* A204120 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                   (* A204121 *)
TableForm[Table[c[n], {n, 1, 10}]]

PARI

up_to = 65703; \\ = binomial(362+1,2)
A204120sq(row,col) = gcd(prime(1+row),prime(1+col));
A204120list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, if(i++ > up_to, return(v)); v[i] = A204120sq((a-(col-1)),col))); (v); };
v204120 = A204120list(up_to);
A204120(n) = v204120[n]; \\ Antti Karttunen, Sep 25 2018