A302345 Primes p such that the set { 1+2p, 1+6p, 1+14p, 1+42p, 1+86p, 1+258p, 1+602p, 1+1806p } does not contain any primes.

67, 97, 127, 163, 307, 317, 337, 349, 409, 521, 523, 547, 643, 709, 757, 811, 839, 857, 919, 967, 997, 1021, 1069, 1087, 1093, 1153, 1277, 1291, 1297, 1301, 1399, 1429, 1459, 1483, 1619, 1627, 1637, 1697, 1709, 1721, 1741, 1789, 1877, 1933, 1949, 1999, 2017, 2029, 2083, 2131, 2179, 2239, 2269, 2311, 2383, 2389, 2437, 2503, 2539, 2557, 2591, 2659, 2671, 2707, 2731

Offset: 1

Author

Max Alekseyev, Apr 05 2018

Keywords

• nonn

Comments

For each term p, the solutions n to the congruence 1^n + 2^n + ... + n^n == p (mod n) form a subset of A014117 U p*A014117. In particular, there are at most 10 solutions for each such p.

The coefficients { 2, 6, 14, 42, 86, 258, 602, 1806 } are the even divisors of 1806 = 2 * 3 * 7 * 43.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcén, Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n), Discrete Applied Mathematics 286 (2020), 3-9. Preprint: arXiv:1602.02407 [math.NT], 2016.

Crossrefs

Solutions to 1^n+2^n+...+n^n == m (mod n): A005408 (m=0), A014117 (m=1), A226960 (m=2), A226961 (m=3), A226962 (m=4), A226963 (m=5), A226964 (m=6), A226965 (m=7), A226966 (m=8), A226967 (m=9), A280041 (m=19), A280043 (m=43), A302343 (m=79), A302344 (m=193).

Programs

Mathematica

Select[Range[3000], PrimeQ[#] && AllTrue[{2, 6, 14, 42, 86, 258, 602, 1806}*# + 1, ! PrimeQ[#1] &] &] (* Amiram Eldar, Aug 09 2020 *)

PARI

{ is_A302345(p) = !vecmax( apply( x->ispseudoprime(1+x*p), 2*divisors(3*7*43) ) ); }