A354428 Primes p such that q divides p^2 + p + 1, r divides q + 1 and p divides r^2 + r + 1 for some primes q and r.

3, 7, 43, 73363, 1477111

Offset: 1

Author

Tomohiro Yamada, May 27 2022

Keywords

• nonn
• more
• hard

Comments

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.

Examples

			43 is a term since 43^2 + 43 + 1 = 3 * 631, 631 + 1 = 2^3 * 79, and 79^2 + 79 + 1 = 3 * 7^2 * 43.
		

Crossrefs

Cf. A101368, A347988.

Cf. A354426 (r corresponding to primes p in this sequence), A354427 (q corresponding to primes p in this sequence).

Programs

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}