A120270 - OEIS

%I #3 Mar 31 2012 13:20:27

%S 1,1,3,1,1,11,1,17,1,1,29,29,1,41,41,4913,17,59,59,1,71,71,1,1,1,1,

%T 101,101,10807,1,1,1,1,137,137,20413,20413,20413,1,1,1,179,1,191,191,

%U 37627,37627,37627,191,43357,227,227,54253,227,1,1,1,269,269,1

%N The numerator of determinant of n X n matrix with elements M[i,j] = 1/(Prime[i] + Prime[j]), i,j=1..n.

%C Many a(n) are equal to 1. It appeares that almost all other a(n) are primes that belong to the Lesser of Twin Primes A001359(k) or equal to the product of two primes from A001359(k), mostly consecutive. a(16) = 17^3 is an exception - it is a cube of a prime from A001359(k). All lesser twin primes from A001359(k) except 5 appear in a(n) for the first time in their natural order. 5 is the only lesser twin prime that does not appear in a(n). If p=Prime[n]>5 is lesser of twin primes then p divides a(n+1).

%F a(n) = numerator[ Det[ 1/(Prime[i] + Prime[j]), {i,1,n},{j,1,n} ]].

%e Matrix begins

%e 1/4 1/5 1/7 1/9 ...

%e 1/5 1/6 1/8 1/10 ...

%e 1/7 1/8 1/10 1/12 ...

%e 1/9 1/10 1/12 1/14 ...

%e ...

%t Numerator[Table[Det[Table[1/(Prime[i]+Prime[j]),{i,1,n},{j,1,n}]],{n,1,60}]]

%Y Cf. A001359.

%K frac,nonn

%O 1,3

%A _Alexander Adamchuk_, Jul 01 2006