This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A216244 #68 Aug 23 2024 04:22:57
%S A216244 4,12,24,60,84,144,180,264,420,480,684,840,924,1104,1404,1740,1860,
%T A216244 2244,2520,2664,3120,3444,3960,4704,5100,5304,5724,5940,6384,8064,
%U A216244 8580,9384,9660,11100,11400,12324,13284,13944,14964,16020,16380,18240,18624,19404,19800
%N A216244 a(n) = (prime(n)^2 - 1)/2 for n >= 2.
%C A216244 Subsequence of A055523 restricted to the case of the other (shorter) leg of the triangle equal to a prime.
%C A216244 There is only one value of a(n) for each prime(n). (This is not necessarily true if the shorter leg is not a prime.)
%C A216244 Note that a(1) is nonexistent since there is no solution with prime = 2.
%C A216244 All terms are divisible by 4.
%C A216244 The values of m (the length of the hypotenuse) always equals a(n) + 1.
%C A216244 a(n) = (prime(n)^2 - 1)/2 for all n > 1.
%C A216244 This follows algebraically given m = a(n) + 1 (or vice versa).
%C A216244 The same two relationships apply when the shorter leg is an odd nonprime, but for only those results corresponding to the longest possible leg of the triangle.
%H A216244 Vincenzo Librandi, <a href="/A216244/b216244.txt">Table of n, a(n) for n = 2..4000</a>
%F A216244 a(n) = (prime(n)^2 - 1)/2 for n >= 2.
%F A216244 a(n) = 4*A061066(n) = A084920(n)/2.
%F A216244 a(n) = A084921(n) for n > 1.
%F A216244 a(n) = (prime(n)-1)*(prime(n)+1)/2 = lcm(prime(n)+1, prime(n)-1) for n > 1 because one of prime(n)+1 or prime(n)-1 is even and the other is divisible by 4. Say prime(n)-1 is divisible by 4; then (prime(n)+1)/2 and (prime(n)-1)/4 must be coprime. - _Frank M Jackson_, Dec 11 2018
%F A216244 Product_{n>=2} (1 + 1/a(n)) = 3/2. - _Amiram Eldar_, Jun 03 2022
%e A216244 24^2 + 7^2 = 625 = 25^2 = (24 +1)^2 and a(4) = (prime(4)^2 -1)/2 = (49 - 1)/2 = 24.
%p A216244 A216244:=n->(ithprime(n)^2-1)/2: seq(A216244(n), n=2..100); # _Wesley Ivan Hurt_, May 03 2017
%t A216244 Table[(Prime[n]^2 - 1)/2, {n, 2, 100}] (* _Vincenzo Librandi_, Jun 15 2014 *)
%o A216244 (PARI) vector(50, n, n++; (prime(n)^2 -1)/2) \\ _G. C. Greubel_, Dec 14 2018
%o A216244 (Magma) [(NthPrime(n)^2 - 1)/2: n in [2..50]]; // _G. C. Greubel_, Dec 14 2018
%o A216244 (Sage) [(nth_prime(n)^2 -1)/2 for n in (2..50)] # _G. C. Greubel_, Dec 14 2018
%Y A216244 Subset of A055523.
%Y A216244 Equals 4*A061066.
%Y A216244 Equals A084921 excluding its first term.
%K A216244 nonn,easy
%O A216244 2,1
%A A216244 _Richard R. Forberg_, May 28 2013
%E A216244 New name (taken from Formula entry) from _Jon E. Schoenfield_, Jul 11 2021