A338566 - OEIS

A338566

Primes p such that (p*q) mod r is prime, where q and r are the next primes after p.

5, 11, 19, 29, 43, 47, 283

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OFFSET

1,1

COMMENTS

a(8) > 40000000 if it exists.

Note that p*q == (r-p)*(r-q) (mod r). As soon as the prime gap grows slow enough, for all large enough p we have (p*q) mod r = (r-p)*(r-q), which is composite, implying finiteness of this sequence. In particular, finiteness would follow from Cramer's conjecture. - Max Alekseyev, Nov 09 2024

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

a(3)=19 is in the sequence because it is prime, the next two primes are 23 and 29, and (19*23) mod 29 = 2, which is prime.

MAPLE

R:= NULL: q:= 2: r:= 3:

count:= 0:

for i from 1 to 10000 do

p:= q; q:= r; r:= nextprime(r);

if isprime(p*q mod r) then count:= count+1; R:= R, p fi

od:

CROSSREFS