A304688 Primes p > 5 such that there is a polygonal number P_s(k) (with s >= 3, k >= 5) equal to p-1.

29, 37, 67, 71, 79, 97, 101, 113, 127, 137, 149, 157, 191, 197, 211, 233, 239, 257, 277, 281, 307, 317, 331, 337, 367, 373, 379, 397, 401, 409, 449, 457, 461, 487, 491, 541, 547, 569, 577, 607, 617, 631, 641, 653, 659, 673, 677, 701, 709, 727, 739, 743, 751, 757, 787, 821, 827, 853, 857

Offset

1,1

Comments

For all primes p > 5, at least one polygonal number exists with P_s(k) = p - 1 when k = 3 or 4, dependent on p mod 6; this is why the sequence is defined for k >= 5.

For s = {14,17} no such P_s(k) exists, since P_14(k) + 1 and P_17(k) + 1 are composites.

For k = 4*m + 1, m > 0 all the numbers P_s(k) + 1 are even and > 2, so they cannot be prime.

For s = 2*m, m >= 2, k = 2*j + 1, j >= 1 all the numbers P_s(k) + 1 are even and > 2, so they cannot be prime.

Links

Table of n, a(n) for n=1..59.

OEIS, Nontrivial polygonal numbers

Example

a(1)-1 = 29-1 = 28 = P_3(7);
a(2)-1 = 37-1 = 36 = P_3(8) = P_4(6).

Mathematica

lst = {}; Do[
If[Resolve[
   Exists[{s, k},
    Prime[m] == 1/2 k (4 + k (-2 + s) - s) + 1 && s >= 3 && k >= 5],
   Integers], lst = Union[lst, {Prime[m]}]], {m, 4, 150}]; lst

Prog

(PARI) isok(p) = {if ((p > 5) && isprime(p), for (s=3, p, if (ispolygonal(p-1, s, &k) && (k>=5), return (1)); ); ); return (0); } \\ Michel Marcus, May 18 2018

Crossrefs

Cf. A000040 (primes), A304690.

Sequence in context: A107134 A139851 A139895 * A341174 A167470 A152865

Adjacent sequences: A304685 A304686 A304687 * A304689 A304690 A304691

Keyword

nonn

Author

Ralf Steiner, May 17 2018

Status

approved