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

%I A204120 #11 Sep 25 2018 20:50:44
%S A204120 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,
%T A204120 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,
%U A204120 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
%N A204120 Symmetric matrix based on f(i,j) = gcd(prime(i+1),prime(j+1)), by antidiagonals.
%C A204120 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.
%C A204120 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
%H A204120 Antti Karttunen, <a href="/A204120/b204120.txt">Table of n, a(n) for n = 1..65703 (the first 362 antidiagonals of array)</a>
%e A204120 Northwest corner:
%e A204120 3 1 1 1
%e A204120 1 5 1 1
%e A204120 1 1 7 1
%e A204120 1 1 1 11
%t A204120 f[i_, j_] := GCD[Prime[i + 1], Prime[j + 1]];
%t A204120 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204120 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204120 Flatten[Table[f[i, n + 1 - i],
%t A204120   {n, 1, 15}, {i, 1, n}]]    (* A204120 *)
%t A204120 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204120 c[n_] := CoefficientList[p[n], x]
%t A204120 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204120 Table[c[n], {n, 1, 12}]
%t A204120 Flatten[%]                   (* A204121 *)
%t A204120 TableForm[Table[c[n], {n, 1, 10}]]
%o A204120 (PARI)
%o A204120 up_to = 65703; \\ = binomial(362+1,2)
%o A204120 A204120sq(row,col) = gcd(prime(1+row),prime(1+col));
%o A204120 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); };
%o A204120 v204120 = A204120list(up_to);
%o A204120 A204120(n) = v204120[n]; \\ _Antti Karttunen_, Sep 25 2018
%Y A204120 Cf. A065091 (main diagonal), A204118, A204121, A204016, A202453.
%K A204120 nonn,tabl
%O A204120 1,1
%A A204120 _Clark Kimberling_, Jan 11 2012