A337818 Odd integers k>=3 such that k*(k-1)/2 divides 2^((k-1)/2)+1.

3, 19, 163, 1459, 370387, 6381667, 30001267, 40417219, 42384547, 42633379, 86093443, 190008019, 268435459, 634471219, 1630068787, 2415919123, 3103616899, 4677743683, 7734924739, 7920392707, 8114552947, 10323768979, 13086951139, 13984274323, 18839387107, 19764019603, 36164859427

Offset: 1

Author

Benoit Cloitre, Sep 23 2020

Keywords

• nonn

Comments

Computed terms are prime. Is it always the case? If not it would be interesting to compute the pseudoprimes.

Conjecture: a(n) == 1 mod 162 for n >= 6. The next few larger terms of the form 162*k+1 are: 44165935747, 46696027123, 85683674179, 88567070707, 101297654083, 131264938963, 131315541283, 177876176419, 195689448883, 196838306227, 213339588643, 310256425603, 378897238243, 502106519683, 588454831747, 611537689459, 641551809187, 735075731107, 745956214867, 755236606483, 771153067603, 872146803043 (and they are all primes). - Chai Wah Wu, Oct 06 2020

The conjecture fails, as a(30) = 48269073907 (prime). - Bill McEachen, Jul 11 2025

Links

• Bill McEachen, Table of n, a(n) for n = 1..32

Programs

Mathematica

Select[Range[3, 400000, 2], PowerMod[2, (# - 1)/2, (t = #*(# - 1)/2)] == t - 1 &] (* Amiram Eldar, Sep 23 2020 *)

PARI

is(n) = n%2 && n>=3 && Mod(2, n*(n-1)/2)^((n-1)/2) == -1 \\ David A. Corneth, Sep 23 2020

Extensions

a(7)-a(27) from Amiram Eldar, Sep 23 2020