A338567 -id:A338567 - 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;

A338570 Primes p such that q*r mod p is prime, where q is the prime preceding p and r is the prime following p.

Original entry on oeis.org

11, 13, 19, 29, 31, 37, 47, 53, 59, 67, 73, 83, 89, 109, 127, 131, 151, 163, 173, 179, 211, 239, 251, 263, 269, 283, 307, 337, 359, 373, 383, 421, 433, 443, 449, 467, 479, 499, 503, 523, 541, 547, 569, 593, 599, 607, 653, 659, 677, 757, 787, 797, 829, 853, 877, 907, 919, 947, 967, 971, 977, 1033

Offset: 1

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

nonn

Primes p such that -A049711(p)*A013632(p) mod p is prime.

Includes primes p such that p-8, p-2 and p+4 are also prime. Dickson's conjecture implies that there are infinitely many of these.

			a(3) = 19 is a member because 19 is prime, the previous and following primes are 17 and 23, and (17*23) mod 19 = 11 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A013632, A049711, A338566, A338567.

R:= NULL: p:= 0: q:= 2: r:= 3:
count:= 0:
while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  if isprime(p*r mod q) then count:= count+1; R:= R, q;  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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A338570 Primes p such that q*r mod p is prime, where q is the prime preceding p and r is the prime following p.

Views

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Keywords

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Examples

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Maple

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

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Python