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

1,1

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.

Mathematica

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

Prog

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

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).

Sequence in context: A256176 A158848 A232634 * A276307 A260806 A180557

Adjacent sequences: A302342 A302343 A302344 * A302346 A302347 A302348

Keyword

nonn

Author

Max Alekseyev, Apr 05 2018

Status

approved