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A298299 a(n) is the smallest b > 0 such that b^(2n) + 1 has all prime divisors p == 1 (mod 2n).

%I A298299 #46 Jul 01 2022 21:36:51
%S A298299 2,2,6,2,10,6,14,2,36,20,66,18,26,28,120,2,170,6,570,140,2184,88,184,
%T A298299 42,110,312,1440,42,116,9060,124,2,7656,34,13650,132,74,228,7800,40,
%U A298299 1066,4158,430,132,6283590,46,94,12,1246,1960
%N A298299 a(n) is the smallest b > 0 such that b^(2n) + 1 has all prime divisors p == 1 (mod 2n).
%C A298299 All the terms are even.
%C A298299 The number a(n)^(2n) + 1 has all divisors d == 1 (mod 2n).
%C A298299 Conjecture: a(n) exists for every n. This is implied by the generalized Bunyakovsky conjecture (Schinzel's hypothesis H).
%C A298299 Theorem: a(n) = 2 if and only if n is a power of 2.
%C A298299 Note: rad(2n) divides rad(a(n)), where rad(m) = A007947(m).
%C A298299 Even numbers 2n such that a(n) = rad(2n) are powers of two and 6, 10, 12, 14, 26, 36, ... Are there infinitely many such numbers?
%C A298299 We have a(n) = 2n = 2, 6, 10, 14, 20, 26, 28, ...
%C A298299 Problem: are there infinitely many even numbers m <> 2^k such that the number m^m + 1 has all divisors d == 1 (mod m)?
%C A298299 From _Kevin P. Thompson_, Mar 13 2022: (Start)
%C A298299 Additional terms: a(46) = 46, a(47) = 94, a(48) = 12, a(49) = 1246, a(50) = 1960, a(52) = 208, a(53) = 636, a(55) = 17600, a(56) = 476.
%C A298299 a(45) > 1000000 (sequence A298398 likewise has a very large value for n=45).
%C A298299 a(51) >= 16524 (a C241 remains to be factored to verify b=16524).
%C A298299 a(54) >= 6864 (a C201 remains to be factored to verify b=6864).
%C A298299 (End)
%H A298299 Kevin P. Thompson, <a href="/A298299/a298299.txt">Factorizations to support known terms of a(n) for n = 1..56</a>
%F A298299 a(n) = min{b > 1: for all prime p, if p | (b^(2n) + 1) then p == 1 (mod 2n)}.
%e A298299 a(5) = 10, because 10^10 + 1 = 10000000001 = 101*3541*27961 and all the prime factors p == 1 (mod 2*5), so all divisors d == 1 (mod 10).
%o A298299 (PARI)
%o A298299 find_a_ORDOWSKI2(n=2, a=1, B_START=2, LIM=10^11,DEBUG=1)={
%o A298299   my(B,FF,LL);
%o A298299   my(fn="_THOMAS_ORDOWSKI_b_a_n.txt");
%o A298299   LL=R2('b,a,n);   \\ R(b,a,n)=(b^n+a)
%o A298299   FF=factor(LL);
%o A298299   if(DEBUG==1,
%o A298299     print(FF);
%o A298299     print(LL);
%o A298299   );
%o A298299   if(Mod(n,2)==0,  \\ n-EVEN
%o A298299     B=FIND_BASE(n,BSTART=B_START,LIM,STEP=2,FF);
%o A298299   );
%o A298299   if(B>0,
%o A298299      return([n,B,[subst(FF,'b,B)]]);
%o A298299   );
%o A298299   return(0);
%o A298299 }
%Y A298299 Cf. A007947, A298076 (see PARI subroutines used for a(48)), A298398.
%K A298299 nonn,hard,more
%O A298299 1,1
%A A298299 _Krzysztof Ziemak_ and _Thomas Ordowski_, Jan 16 2018
%E A298299 a(31)-a(44) from _Kevin P. Thompson_, Mar 13 2022
%E A298299 a(45)-a(50) from _Daniel Suteu_, Jul 01 2022