A283562 Primes of the form (p^2 - q^2) / 24 with primes p > q > 3.

2, 3, 5, 7, 11, 13, 17, 23, 37, 43, 47, 53, 67, 73, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 223, 233, 257, 263, 277, 283, 293, 313, 337, 347, 353, 373, 397, 433, 443, 467, 487, 523, 547, 563, 577, 593, 607, 613, 617, 643, 647, 653, 733, 743, 757, 773, 787, 797, 887, 907, 937, 947, 953, 977

Offset

1,1

Comments

Note that p - q must be <= 12. Also note that there can be corresponding prime pairs (q, p) more than one way, i.e., (7, 13), (13, 17), (29, 31): (13^2 - 7^2)/24 = (17^2 - 13^2)/24 = (31^2 - 29^2)/24 = 5.

There are no terms of A045468 > 11.

Union of {2}, A006489, A060212, A092110, and A125272. - Robert Israel, Mar 13 2017

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

For n > 5, a(n) == {3,7} mod 10.

Example

3 is a term since (11^2 - 7^2)/24 = 3 and 3, 7, 11 are prime numbers.

Maple

select(r -> isprime(r) and ((isprime(3*r+2) and isprime(3*r-2))
  or (isprime(6*r+1) and isprime(6*r-1))
  or (isprime(2*r+3) and isprime(2*r-3))
or (isprime(r+6) and isprime(r-6))), [2, seq(i, i=3..1000, 2)]); # Robert Israel, Mar 13 2017

Mathematica

ok[n_] := PrimeQ[n] && Block[{p, q, s = Reduce[p^2-q^2 == 24 n && p>3 && q>3, {p, q}, Integers]}, If[s === {}, False, Or @@ And @@@ PrimeQ[{p, q} /. List@ ToRules@s]]]; Select[Range@1000, ok] (* Giovanni Resta, Mar 11 2017 *)

Prog

(PARI) isA124865(n) = if(n%24, isprimepower(n+4)==2 || isprimepower(n+9)==2, fordiv(n/4, d, if(isprime(n/d/4+d) && isprime(n/d/4-d), return(1))); 0)
lista(nn) = forprime(p=2, nn, if(isA124865(24*p), print1(p", ")))

Crossrefs

Cf. A006489, A060212, A092110, A124865, A124866, A125272, A283532.

Sequence in context: A216823 A293199 A268812 * A152245 A007693 A174048

Adjacent sequences: A283559 A283560 A283561 * A283563 A283564 A283565

Keyword

nonn

Author

Altug Alkan and Thomas Ordowski, Mar 11 2017

Status

approved