cp's OEIS Frontend

A283680 Odd numbers of the form (p^2 - q^2) / 24 with primes p > q > 3.

%I A283680 #29 Apr 06 2017 21:28:32
%S A283680 1,3,5,7,11,13,15,17,21,23,25,27,33,35,37,39,43,45,47,53,55,57,63,65,
%T A283680 67,69,73,75,77,85,87,91,93,95,97,103,105,107,113,115,117,123,127,133,
%U A283680 135,137,143,145,147,153,155,157,163,165,167,173,175,177,183,185,187,193,195,203,205,207,209,213,215,217
%N A283680 Odd numbers of the form (p^2 - q^2) / 24 with primes p > q > 3.
%C A283680 If p == 3(mod 8) and p == 1(mod 3), and if q == 7(mod 8) and q == 1(mod 3); then (p^2 - q^2)/24 is odd. Thus, this sequence is infinite.
%C A283680 Note: if p - q > 12, then (p^2 - q^2)/24 is composite.
%C A283680 Theorem: (p^2 - q^2)/24 is an odd integer if and only if pq == +-3(mod 8). - Carl Pomerance, Mar 14 2017
%C A283680 The complement is: 9, 19, 29, 31, 41, 49, 51, 59, 61, 71, 79, 81, 83, 89, 99, 101, 109, 111, 119, 121, 125, 129, 131, 139, 141, 149, 151, 159, 161, 169, 171, ... - _Robert G. Wilson v_, Mar 14 2017
%H A283680 Robert G. Wilson v, <a href="/A283680/b283680.txt">Table of n, a(n) for n = 1..10000</a>
%F A283680 a(n) ~ cn(loglog n)^(3/2)/(log n)^(1+eta), where eta = 1-(1+loglog 2)/log 2 = 0.086..., perhaps with c about 4/3. - Conjectured by Carl Pomerance, Mar 15 2017
%t A283680 p = 7; lst = {}; While[p < 501, q = 5; While[q < p, If[Mod[p^2 - q^2, 24] == 0 && Mod[(p^2 - q^2)/24, 2] == 1, AppendTo[lst, (p^2 - q^2)/24]; lst = Union@ lst]; q = NextPrime@ q];  p = NextPrime@ p]; Take[lst, 75] (* _Robert G. Wilson v_, Mar 14 2017 *)
%Y A283680 Cf. A283562.
%K A283680 nonn
%O A283680 1,2
%A A283680 _Thomas Ordowski_ and _Altug Alkan_, Mar 14 2017