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A345213 Primes p such that q^r == r^q (mod p), where p,q,r are consecutive primes.

%I A345213 #11 Jun 11 2021 12:23:07
%S A345213 2,11,29,59,71,149,269,431,569,599,727,1031,1061,1229,1289,1319,1451,
%T A345213 1619,2129,2339,2381,2549,2711,2789,3299,3539,4019,4049,4091,4649,
%U A345213 4721,5099,5441,5519,5639,5741,5849,6269,6359,6569,6701,6959,7211,8009,8999,9041,9341,10091,10859,11489,11831
%N A345213 Primes p such that q^r == r^q (mod p), where p,q,r are consecutive primes.
%C A345213 Terms not in A049437 include 2, 11, 727 and 22571. Are there others?
%C A345213 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?
%H A345213 Robert Israel, <a href="/A345213/b345213.txt">Table of n, a(n) for n = 1..10000</a>
%e A345213 a(3) = 29 is a term because 29, 31 and 37 are consecutive primes and 37^31 == 31^37 == 19 (mod 29).
%p A345213 q:= 2: r:= 3: R:= NULL: count:= 0:
%p A345213 while count < 100 do
%p A345213   p:= q; q:= r; r:= nextprime(r);
%p A345213   if q&^r - r&^q mod p = 0 then count:= count+1; R:= R, p fi
%p A345213 od:
%p A345213 R;
%Y A345213 Includes A049437 except for 3.
%K A345213 nonn
%O A345213 1,1
%A A345213 _J. M. Bergot_ and _Robert Israel_, Jun 10 2021