login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A216244
a(n) = (prime(n)^2 - 1)/2 for n >= 2.
6
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
2,1
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.
LINKS
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
EXAMPLE
24^2 + 7^2 = 625 = 25^2 = (24 +1)^2 and a(4) = (prime(4)^2 -1)/2 = (49 - 1)/2 = 24.
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 *)
PROG
(PARI) vector(50, n, n++; (prime(n)^2 -1)/2) \\ G. C. Greubel, Dec 14 2018
(Magma) [(NthPrime(n)^2 - 1)/2: n in [2..50]]; // G. C. Greubel, Dec 14 2018
(Sage) [(nth_prime(n)^2 -1)/2 for n in (2..50)] # G. C. Greubel, Dec 14 2018
CROSSREFS
Subset of A055523.
Equals 4*A061066.
Equals A084921 excluding its first term.
Sequence in context: A003203 A051193 A356550 * A215223 A318610 A296358
KEYWORD
nonn,easy
AUTHOR
Richard R. Forberg, May 28 2013
EXTENSIONS
New name (taken from Formula entry) from Jon E. Schoenfield, Jul 11 2021
STATUS
approved