A061066 -id:A061066 - cp's OEIS frontend

A216244 a(n) = (prime(n)^2 - 1)/2 for n >= 2.

4, 12, 24, 60, 84, 144, 180, 264, 420, 480, 684, 840, 924, 1104, 1404, 1740, 1860, 2244, 2520, 2664, 3120, 3444, 3960, 4704, 5100, 5304, 5724, 5940, 6384, 8064, 8580, 9384, 9660, 11100, 11400, 12324, 13284, 13944, 14964, 16020, 16380, 18240, 18624, 19404, 19800

Offset: 2

Richard R. Forberg, May 28 2013

Subsequence of A055523 restricted to the case of the other (shorter) leg of the triangle equal to a prime.
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.)
Note that a(1) is nonexistent since there is no solution with prime = 2.
All terms are divisible by 4.
The values of m (the length of the hypotenuse) always equals a(n) + 1.
a(n) = (prime(n)^2 - 1)/2 for all n > 1.
This follows algebraically given m = a(n) + 1 (or vice versa).
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.

			24^2 + 7^2 = 625 = 25^2 = (24 +1)^2  and a(4) = (prime(4)^2 -1)/2 = (49 - 1)/2 = 24.

Vincenzo Librandi, Table of n, a(n) for n = 2..4000

Subset of A055523.
Equals 4*A061066.
Equals A084921 excluding its first term.

[(NthPrime(n)^2 - 1)/2: n in [2..50]]; // G. C. Greubel, Dec 14 2018

A216244:=n->(ithprime(n)^2-1)/2: seq(A216244(n), n=2..100); # Wesley Ivan Hurt, May 03 2017

Table[(Prime[n]^2 - 1)/2, {n, 2, 100}] (* Vincenzo Librandi, Jun 15 2014 *)

vector(50, n, n++; (prime(n)^2 -1)/2) \\ G. C. Greubel, Dec 14 2018

[(nth_prime(n)^2 -1)/2 for n in (2..50)] # G. C. Greubel, Dec 14 2018

a(n) = (prime(n)^2 - 1)/2 for n >= 2.
a(n) = 4*A061066(n) = A084920(n)/2.
a(n) = A084921(n) for n > 1.
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
Product_{n>=2} (1 + 1/a(n)) = 3/2. - Amiram Eldar, Jun 03 2022

New name (taken from Formula entry) from Jon E. Schoenfield, Jul 11 2021

A024700 a(n) = (prime(n+2)^2 - 1)/3.

Original entry on oeis.org

8, 16, 40, 56, 96, 120, 176, 280, 320, 456, 560, 616, 736, 936, 1160, 1240, 1496, 1680, 1776, 2080, 2296, 2640, 3136, 3400, 3536, 3816, 3960, 4256, 5376, 5720, 6256, 6440, 7400, 7600, 8216, 8856, 9296, 9976, 10680, 10920, 12160, 12416, 12936, 13200, 14840, 16576, 17176

Offset: 1

Clark Kimberling, Dec 11 1999

nonn

Numbers of the form 4*h*(3*h +- 1). - Vincenzo Librandi, May 21 2013

This sequence is also: Numbers n such that k is prime and its square is of the form 3*n + 1 (i.e., k^2 = 3*n + 1). For this case, the sequence is to be prepended with a(0) = 1. - G. C. Greubel, Sep 18 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040 (prime(n)), A001248, A024701, A024702, A061066, A081115, A084920, A084921, A084922.

[(NthPrime(n+2)^2-1)/3: n in [1..50]]; // Bruno Berselli, May 22 2013

Select[Range[2,10000], PrimeQ[Sqrt[3*#+1]] &] (* G. C. Greubel, Sep 18 2016 *)
(Prime[Range[3,50]]^2-1)/3 (* Harvey P. Dale, May 05 2022 *)

a(n) = (prime(n+2)^2-1)/3; \\ Altug Alkan, Sep 18 2016

[(n^2 -1)/3 for n in prime_range(4,301)] # G. C. Greubel, May 02 2024

a(n) = (A001248(n+2) - 1)/3. - Elmo R. Oliveira, Jan 20 2023

a(n) = 8*A024702(n+2) = 4*A081115(n+2) = 2*A084922(n+2) = (2/3)*A084921(n) = (4/3)*A024701(n+1) = (8/3)*A061066(n+2). - Alois P. Heinz, Jan 20 2023

cp's OEIS Frontend

A216244 a(n) = (prime(n)^2 - 1)/2 for n >= 2.

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