A354426 - OEIS

%I #11 Aug 11 2024 23:45:34

%S 2,7,79,5569,9829

%N Primes p such that q divides p^2 + p + 1, r divides q^2 + q + 1 and p divides r + 1 for some primes q and r.

%C There are no other terms below 2^24.

%C There are no other terms below 10^8. - _Lucas A. Brown_, Aug 11 2024

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A354426.py">Python program</a>.

%H Tomohiro Yamada, <a href="https://doi.org/10.2140/moscow.2021.10.249">On a problem of De Koninck</a>, Moscow Journal of Combinatorics and Number Theory, 10:3 (2021), 249-260, correction, 10:4 (2021), 339.

%e 7 is a term since 7^2 + 7 + 1 = 3 * 19, 3^2 + 3 + 1 = 13 and 13 + 1 = 2 * 7.

%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+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, A347988.

%K nonn,more,hard

%O 1,1

%A _Tomohiro Yamada_, May 27 2022