A303379 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).

163, 223, 439, 499, 613, 691, 733, 769, 919, 1009, 1039, 1123, 1249, 1423, 1459, 1543, 1579, 1693, 1741, 1753, 1759, 2179, 2203, 2251, 2281, 2293, 2371, 2503, 2539, 2671, 2683, 2749, 2833, 2953, 3019, 3163, 3319, 3343, 3499, 3739, 3793, 3943, 4099, 4363, 4513, 4603

Offset

1,1

Comments

The term r*(r + 1)/2 is the triangular number T(r).

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.

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).

Links

Table of n, a(n) for n=1..46.

Mathematica

lst = {}; Do[p = Prime[n]; f = False;
If[1 == Mod[p, 6], f = True;
  Do[If[r == Mod[p - 2, 1/2 r (1 + r)], f = False], {r, 5, p - 3}]];
If[f && ! PrimeQ[p - 2], lst = AppendTo[lst, p]], {n, 1, 500}]; lst

Crossrefs

Cf. A002476, A316188, A006512, A000217.

Sequence in context: A272984 A230869 A132250 * A142940 A092529 A142373

Adjacent sequences: A303376 A303377 A303378 * A303380 A303381 A303382

Keyword

nonn

Author

Ralf Steiner, Apr 22 2018

Status

approved