A355298 - OEIS

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.

3, 13, 61, 127, 399403 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	COMMENTS	`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.`
	LINKS	`Table of n, a(n) for n=1..5.` `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.`
	EXAMPLE	`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.`
	PROG	`(PARI) 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}`
	CROSSREFS	`Cf. A101368, A347988, A354427.` `Sequence in context: A106884 A232611 A238445 * A328704 A341077 A357749` `Adjacent sequences: A355295 A355296 A355297 * A355299 A355300 A355301`
	KEYWORD	`nonn,more,hard`
	AUTHOR	`Tomohiro Yamada, Jun 28 2022`
	STATUS	`approved`

Last modified August 30 00:57 EDT 2024. Contains 375520 sequences. (Running on oeis4.)