A124865 Numbers of the form p^2-q^2 with p > q prime.

5, 16, 21, 24, 40, 45, 48, 72, 96, 112, 117, 120, 144, 160, 165, 168, 192, 240, 264, 280, 285, 288, 312, 336, 352, 357, 360, 408, 432, 480, 504, 520, 525, 528, 552, 600, 648, 672, 720, 768, 792, 816, 832, 837, 840, 888, 912, 936, 952, 957, 960, 1008, 1032, 1080

Offset

1,1

Comments

The only prime term is a(1) = 5. All odd terms are of the form p^2-4. All even terms are divisible by 8. Numbers of the form (p^2-q^2)/8 (p, q odd primes, p>q) are listed in A124866(n) = {2,3,5,6,9,12,14,15,18,20,21,24,30,33,35,36,39,42,44,45,51,54,60,63,65,66,69, 75,81,84,90,96,99,...}.

Elliott & Richner call these "ans numbers". - Charles R Greathouse IV, Feb 17 2014

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

N. E. Elliott and D. Richner, An investigation of the set of ans numbers, Missouri J. of Math. Sci. 15 (2003), pp. 189-199.

Florian Luca, On the densities of some subsets of integers, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.

Formula

a(n) >> n log n, see Luca. - Charles R Greathouse IV, Feb 17 2014

Mathematica

With[{nn=60}, Take[Union[#[[2]]^2-#[[1]]^2&/@Subsets[Prime[Range[nn]], {2}]], nn]] (* Harvey P. Dale, Aug 21 2015 *)

Prog

(PARI) is(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) \\ Charles R Greathouse IV, Feb 17 2014

Crossrefs

Apart from a(1), a subsequence of A177713.

Cf. A045636 Numbers of the form p^2+q^2, p, q primes. Cf. A124866 Numbers of the form (p^2-q^2)/8, p, q odd primes, p>q.

Sequence in context: A057281 A204920 A168469 * A090781 A191264 A077469

Adjacent sequences: A124862 A124863 A124864 * A124866 A124867 A124868

Keyword

nonn

Author

Alexander Adamchuk, Nov 10 2006

Status

approved