%I #7 Oct 19 2019 23:41:10
%S 5,12,13,14,23,25,27,31,34,38,40,41,42,44,57,58,59,61,63,64,65,80,82,
%T 84,85,86,88,90,94,96,97,100,101,103,107,109,111,113,115,117,119,138,
%U 139,140,142,144,145,146,148,150,151,152,173,175,176,177,179,181,183
%N Map positive rational numbers to positive integers by diagonal method using c(p,q) = (p + q - 2) * (p + q - 1) / 2 + p where p and q are positive integers. a(n) is an increasing sequence including all c(p,q) where gcd(p,q) > 1.
%e a(1) = c(2,2) = (2 + 2 - 2) * (2 + 2 - 1) / 2 + 2 = 5 because gcd(2,2) = 2 > 1.
%e a(2) = c(2,4) = (2 + 4 - 2) * (2 + 4 - 1) / 2 + 2 = 12 because gcd(2,4) = 2 > 1.
%o (PARI) a(n)=if(n<1,0,n=a(n-1); until(1<gcd(n-binomial(floor(1/2+sqrt(2 *n)),2),binomial(floor(3/2+sqrt(2*n)),2)-n+1),n++); n)
%K nonn
%O 1,1
%A Robert A. Stump (bee_ess107(AT)msn.com), Oct 18 2002