A354427 -id:A354427 - cp's OEIS frontend

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

Tomohiro Yamada, May 27 2022

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.

			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.

Cf. A101368, A347988.

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

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}

A355298 Primes p such that q divides p + 1, r divides q^2 + q + 1, s divides r^2 + r + 1, and p divides s^2 + s + 1 for some primes q, r, and s.

Original entry on oeis.org

3, 13, 61, 127, 399403

Offset: 1

Tomohiro Yamada, Jun 28 2022

There are no other terms below 2^24.

If rad(n)^2 = sigma(n), where rad(n) = A007927(n) is the largest squarefree number dividing n and sigma(n) = A000203(n) is the sum of divisors of n, and there exists just one odd prime factor p dividing n exactly once, then p must belong to A354427 or this sequence.

			61 is a term since 61 + 1 = 2 * 31, 31^2 + 31 + 1 = 3 * 331, 3^2 + 3 + 1 = 13, and 13^2 + 13 + 1 = 3 * 61.

Tomohiro Yamada, On a problem of De Koninck, Moscow Journal of Combinatorics and Number Theory, 10:3 (2021), 249-260, correction, 10:4 (2021), 339.

Cf. A101368, A347988, A354427.

is(p)={my(W, V1, V2, V3, V4, q1, q2, q3, q4, i1, i2, i3, i4, l1, l2, l3, l4); W=0; V1=factor(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]; V4=factor(q3^2+q3+1); l4=length(V4[, 1]); for(i4=1, l4, q4=V4[i4, 1];if(q4==p, W=[p, q1, q2, q3]))))); W}

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI