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A204120
Symmetric matrix based on f(i,j) = gcd(prime(i+1),prime(j+1)), by antidiagonals.
3
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
1,1
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
EXAMPLE
Northwest corner:
3 1 1 1
1 5 1 1
1 1 7 1
1 1 1 11
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}]]
PROG
(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
CROSSREFS
Cf. A065091 (main diagonal), A204118, A204121, A204016, A202453.
Sequence in context: A123940 A350447 A339969 * A342666 A268032 A339899
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 11 2012
STATUS
approved