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%I #8 Nov 01 2021 00:55:58
%S 3,13,31,43,61,307,331,631,5233
%N Primes p such that q divides p^2 + p + 1, r divides q^2 + q + 1 and p divides r^2 + r + 1 for some primes q and r.
%C There are no other terms below 2^24. Clearly, if a prime p is in this sequence, then so are q and r. Taking such prime triples (p, q, r) with p smallest, we have three triples (3, 13, 61), (31, 331, 5233), and (43, 631, 307) below 2^24.
%e 3 is a term since 3^2 + 3 + 1 = 13, 13^2 + 13 + 1 = 3 * 61, and 61^2 + 61 + 1 = 3 * 13 * 97.
%o (PARI) is(p)={my(W,V1,V2,V3,q1,q2,q3,i1,i2,i3,l1,l2,l3);W=0;V1=factor(p^2+p+1);l1=length(V1[,1]);for(i1=1,l1,q1=V1[i1,1];V2=factor(q1^2+q1+1);l2=length(V2[,1]);for(i2=1,l2,q2=V2[i2,1];V3=factor(q2^2+q2+1);l3=length(V3[,1]);for(i3=1,l3,q3=V3[i3,1];if(q3==p,W=[p,q1,q2]))));W}
%Y Cf. A101368.
%K nonn,hard,more
%O 1,1
%A _Tomohiro Yamada_, Sep 23 2021
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