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Numbers n such that 2n+1 is prime, 2n+3 is square, and 2n+5 is triangular.
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%I #30 Oct 03 2017 06:26:17

%S 83,1103,98123,1275203,1471585499,130674477863,1698208392983,

%T 728880893315750472460743125221814632790855997983,

%U 14557044964961408694308418152479413877871536090878805525883

%N Numbers n such that 2n+1 is prime, 2n+3 is square, and 2n+5 is triangular.

%C The first 5 terms are primes of the form 6k+5.

%C a(17) has 1559 digits and a(18) > 10^3000, if it exists. - _Giovanni Resta_, Sep 15 2017

%e 83 is a term as 2*83+1=167 (prime), 2*83+3=169=(13)^2 (square), and 2*83+5=171 (triangular).

%t Select[Range[10^11], PrimeQ[2#+1] && IntegerQ@Sqrt[2#+3]&& OddQ@Sqrt[8(2#+5)+1]&]

%o (PARI) for(n=1, 10^11, isprime(2*n+1) && issquare(2*n+3)&& ispolygonal(2*n+5,3) && print1(n", "))

%Y Cf. A000040 (primes), A000290 (squares), A000217 (triangular numbers).

%K nonn

%O 1,1

%A _Waldemar Puszkarz_, Sep 14 2017

%E a(6)-a(9) from _Giovanni Resta_, Sep 15 2017