A356471 -id:A356471 - cp's OEIS frontend

A356475 First of three consecutive primes p,q,r such that pq + qr + r*p is prime.

2, 3, 5, 7, 17, 29, 37, 41, 43, 67, 83, 103, 137, 157, 179, 181, 193, 227, 277, 283, 347, 359, 383, 431, 457, 461, 607, 661, 701, 709, 757, 773, 823, 827, 839, 859, 937, 967, 1013, 1051, 1061, 1109, 1129, 1187, 1201, 1213, 1249, 1283, 1307, 1327, 1373, 1423, 1439, 1471, 1481, 1487, 1543, 1567

Offset: 1

J. M. Bergot and Robert Israel, Aug 08 2022

nonn

			a(4) = 7 is a term because 7, 11, 13 are three consecutive primes with 7*11 + 11*13 + 13*7 = 311 which is prime.

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

Cf. A189759, A356471, A356477.

R:= NULL: count:= 0:
P:= Vector(3,ithprime):
while count < 100 do
  x:= P[1]*P[2]+P[2]*P[3]+P[3]*P[1];
  if isprime(x) then R:= R, P[1]; count:= count+1 fi;
  P[1..2]:= P[2..3];
  P[3]:= nextprime(P[3]);
od:
R;

Select[Partition[Prime[Range[250]], 3, 1], PrimeQ[Total[# * RotateLeft[#]]] &][[;; , 1]] (* Amiram Eldar, Aug 08 2022 *)

list(lim)=my(v=List(),p=2,q=3); forprime(r=5,nextprime(nextprime(lim\1+1)+1), if(isprime(p*q + q*r + r*p), listput(v,p)); p=q; q=r); Vec(v) \\ Charles R Greathouse IV, Sep 06 2022

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q, r = 2, 3, 5
    while True:
        if isprime(p*q + q*r + r*p): yield p
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 58))) # Michael S. Branicky, Aug 08 2022

A356477 a(n) is the start of the first sequence of 2n+1 consecutive primes p_1, p_2, ..., p_(2n+1) such that p_1p_2 + p_2p_3 + ... + p_(2n)p_(2n+1) + p_(2n+1)*p_1 is prime.

Original entry on oeis.org

2, 19, 19, 2, 23, 2, 7, 7, 2, 5, 113, 5, 29, 13, 67, 53, 11, 11, 5, 23, 7, 43, 5, 2, 31, 73, 13, 3, 89, 5, 11, 3, 89, 31, 43, 2, 37, 2, 23, 7, 11, 19, 43, 23, 5, 2, 23, 3, 29, 5, 17, 3, 31, 29, 53, 29, 7, 13, 73, 3, 5, 43, 29, 17, 5, 37, 19, 11, 71, 7, 2, 43, 13, 19, 2, 59, 7, 29, 113, 13, 5, 11

Offset: 1

J. M. Bergot and Robert Israel, Aug 08 2022

nonn

			a(2) = 19 because 19 is the start of the 2*2+1 = 5 consecutive primes 19, 23, 29, 31, 37 with 19*23 + 23*29 + 29*31 + 31*37 + 37*19 = 3853 prime, and no earlier 5-tuple of consecutive primes works.

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

Cf. A070934, A356471, A356475.

f:= proc(m) local P,x,i,n;
  n:= 2*m+1;
  P:= Vector(n,ithprime);
do
   x:= add(P[i]*P[i+1],i=1..n-1)+P[n]*P[1];
   if isprime(x) then return P[1] fi;
   P[1..n-1]:= P[2..n];
   P[n]:= nextprime(P[n]);
od
end proc:
map(f, [$1..100]);

from sympy import isprime, nextprime, prime, primerange
def a(n):
    p = list(primerange(1, prime(2*n+1)+1))
    while True:
        if isprime(sum(p[i]*p[i+1] for i in range(len(p)-1))+p[-1]*p[0]):
            return p[0]
        p = p[1:] + [nextprime(p[-1])]
print([a(n) for n in range(1, 83)]) # Michael S. Branicky, Aug 08 2022

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A356475 First of three consecutive primes p,q,r such that pq + qr + r*p is prime.

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A356477 a(n) is the start of the first sequence of 2n+1 consecutive primes p_1, p_2, ..., p_(2n+1) such that p_1p_2 + p_2p_3 + ... + p_(2n)p_(2n+1) + p_(2n+1)*p_1 is prime.

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cp's OEIS Frontend

A356475 First of three consecutive primes p,q,r such that p*q + q*r + r*p is prime.

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Maple

Mathematica

PARI

Python

A356477 a(n) is the start of the first sequence of 2*n+1 consecutive primes p_1, p_2, ..., p_(2*n+1) such that p_1*p_2 + p_2*p_3 + ... + p_(2*n)*p_(2*n+1) + p_(2*n+1)*p_1 is prime.

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A356475 First of three consecutive primes p,q,r such that pq + qr + r*p is prime.

A356477 a(n) is the start of the first sequence of 2n+1 consecutive primes p_1, p_2, ..., p_(2n+1) such that p_1p_2 + p_2p_3 + ... + p_(2n)p_(2n+1) + p_(2n+1)*p_1 is prime.