A338578 -id:A338578 - cp's OEIS frontend

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

Offset: 1

J. M. Bergot and Robert Israel, Nov 02 2020

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

			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.

Contained in A338578.

Cf. A338567.

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:
R;

A338577 Primes p such that A013632(p)*A105161(p) > p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 31, 47, 83, 89, 113, 199, 1327

Offset: 1

Robert Israel, Nov 03 2020

Primes p such that (q-p)*(r-p) > p, where q and r are the next two primes after p.
a(16) > 10^8 if it exists.
Sequence is finite if Cramér's conjecture is true. - Chai Wah Wu, Nov 03 2020
Data from A002386 and A005250 show that a(16) > 18361375334787046697 if it exists. - Jason Yuen, Jun 13 2024

			a(5) = 11 is a member because 11 is prime, the next two primes are 13 and 17, and (13-11)*(17-11) = 12 > 11.

Contains A338567.
Cf. A013632, A105161, A338578.
Cf. A002386, A005250.

p:= 0: q:=2:r:= 3: R:= NULL:
while p < 10^4 do
  p:= q: q:= r: r:= nextprime(r);
  if (q-p)*(r-p) > p then R:= R, p fi
od:
R;

from sympy import nextprime
A338577_list, p, q, r = [], 2,3,5
while p < 10**6:
    if (q-p)*(r-p) > p:
        A338577_list.append(p)
    p, q, r = q, r, nextprime(r) # Chai Wah Wu, Nov 03 2020

cp's OEIS Frontend

A338566 Primes p such that (p*q) mod r is prime, where q and r are the next primes after p.

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