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%I #69 Jul 03 2018 18:34:20
%S 163,223,439,499,613,691,733,769,919,1009,1039,1123,1249,1423,1459,
%T 1543,1579,1693,1741,1753,1759,2179,2203,2251,2281,2293,2371,2503,
%U 2539,2671,2683,2749,2833,2953,3019,3163,3319,3343,3499,3739,3793,3943,4099,4363,4513,4603
%N Primes p such that p == 1 (mod 6), p-2 is composite, and for all r in the range 5 <= r < p - 2, p-2 is not congruent to r (mod r*(r + 1)/2).
%C The term r*(r + 1)/2 is the triangular number T(r).
%C If p - 2 is composite, a(n) is not the greater member (A006512) p >= 7 of a twin prime, although such p always meet the remaining conditions.
%C The set {a(n)}+{A006512}/{5} is one of four disjunct classes of primes p >= 7 dependent on all cases of the two conditions p == [1 | 5] (mod 6) and if in the range 5 <= r < p - 2 [at least one | not any] r exists with p - 2 == r (mod r*(r + 1)/2).
%t lst = {}; Do[p = Prime[n]; f = False;
%t If[1 == Mod[p, 6], f = True;
%t Do[If[r == Mod[p - 2, 1/2 r (1 + r)], f = False], {r, 5, p - 3}]];
%t If[f && ! PrimeQ[p - 2], lst = AppendTo[lst, p]], {n, 1, 500}]; lst
%Y Cf. A002476, A316188, A006512, A000217.
%K nonn
%O 1,1
%A _Ralf Steiner_, Apr 22 2018
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