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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.
3
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
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
KEYWORD
nonn
AUTHOR
Max Alekseyev, Apr 05 2018
STATUS
approved