login
A345213
Primes p such that q^r == r^q (mod p), where p,q,r are consecutive primes.
1
2, 11, 29, 59, 71, 149, 269, 431, 569, 599, 727, 1031, 1061, 1229, 1289, 1319, 1451, 1619, 2129, 2339, 2381, 2549, 2711, 2789, 3299, 3539, 4019, 4049, 4091, 4649, 4721, 5099, 5441, 5519, 5639, 5741, 5849, 6269, 6359, 6569, 6701, 6959, 7211, 8009, 8999, 9041, 9341, 10091, 10859, 11489, 11831
OFFSET
1,1
COMMENTS
Terms not in A049437 include 2, 11, 727 and 22571. Are there others?
Are there primes p other than 7 such that p^q == q^p (mod r), or primes p other than 41 such that p^r == r^p (mod q), where p,q,r are consecutive primes?
LINKS
EXAMPLE
a(3) = 29 is a term because 29, 31 and 37 are consecutive primes and 37^31 == 31^37 == 19 (mod 29).
MAPLE
q:= 2: r:= 3: R:= NULL: count:= 0:
while count < 100 do
p:= q; q:= r; r:= nextprime(r);
if q&^r - r&^q mod p = 0 then count:= count+1; R:= R, p fi
od:
R;
CROSSREFS
Includes A049437 except for 3.
Sequence in context: A061238 A046500 A062123 * A117560 A024178 A009312
KEYWORD
nonn
AUTHOR
J. M. Bergot and Robert Israel, Jun 10 2021
STATUS
approved