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COMMENTS
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There are no other terms below 2^24.
The first five terms correspond to 7, 2, 79, 9829, and 5569 in A354426 respectively.
Similarly, these correspond to 13, 3 (or 19), 631, 1794067711, and 10855016833 in A354427 respectively.
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PROG
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(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+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}
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