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%I #15 Jul 22 2016 11:21:27
%S 7,24,32,87,104,175,184,287,335,399,552,560,759,840,847,1000,1232,
%T 1287,1455,1504,1719,1824,2015,2232,2320,2464,2992,3047,3080,3160,
%U 3552,3912,3952,4199,4927,4959,5512,5575,5719,5887,6104,6600,7175,7279,7455,8207,8399,8624,8855,8992,9424,9775,9799,10000
%N Numbers n such that n = a*b and 2*n + 1 = c*d such that a + b = c + d.
%H Charles R Greathouse IV, <a href="/A270422/b270422.txt">Table of n, a(n) for n = 1..10000</a>
%e n = 7 = 1*7 and 2*n + 1 = 15 = 3*5 such that 1 + 7 = 3 + 5.
%e n = 24 = 2*12 and 2*n + 1 = 49 = 7*7 such that 2 + 12 = 7 + 7.
%e n = 32 = 2*16 and 2*n + 1 = 65 = 5*13 such that 2 + 16 = 5 + 13.
%e n = 87 = 3*29 and 2*n + 1 = 175 = 7*25 such that 3 + 29 = 7 + 25.
%t sd[n_] := Plus @@@ ({#, n/#} & /@ Select[ Divisors@ n, #^2 <= n &]); Select[Range@ 10000, {} != Intersection[ sd[#], sd[2*# + 1]] &] (* _Giovanni Resta_, Jul 12 2016 *)
%o (PARI) is(n)=my(m=2*n+1,d=divisors(n),e=divisors(m)); for(i=1,#d, for(j=1,#e, if(d[i] + n/d[i] == e[j] + m/e[j], return(1)))); 0 \\ _Charles R Greathouse IV_, Jul 21 2016
%o (PARI) is(n)=my(m=2*n+1,d=divisors(m),t,s); for(i=1,#d, t=d[i]+m/d[i]; if(issquare(t^2 - 4*n, &s) && (t+s)*(t-s)==4*n && (t+s)%2==0, return(1))); 0 \\ _Charles R Greathouse IV_, Jul 21 2016
%K nonn
%O 1,1
%A _Debapriyay Mukhopadhyay_, Jul 12 2016
%E Missing term 8992 from _Giovanni Resta_, Jul 12 2016
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