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

Author

Richard R. Forberg, May 28 2013

Keywords

• nonn
• easy

Comments

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.

Examples

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

Links

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

Crossrefs

Subset of A055523.

Equals 4*A061066.

Equals A084921 excluding its first term.

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

Sage

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

Formula

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

Extensions

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