

A216244


Numbers a(n) such that a(n)^2 + prime(n)^2 = m^2 for some integer m.


4



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
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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

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


FORMULA

a(n) = (prime(n)^2 1)/2 for n > 1.
a(n) = 4*A061066(n).
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


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)^21)/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: A136486 A003203 A051193 * A215223 A318610 A296358
Adjacent sequences: A216241 A216242 A216243 * A216245 A216246 A216247


KEYWORD

nonn,easy


AUTHOR

Richard R. Forberg, May 28 2013


EXTENSIONS

More terms from Vincenzo Librandi, Jun 15 2014


STATUS

approved



