A356475 -id:A356475 - cp's OEIS frontend

A189759 Numbers pqr such that pq + pr + qr is prime, where p, q, and r are primes.

30, 42, 66, 70, 78, 105, 114, 130, 154, 165, 174, 182, 222, 231, 238, 246, 255, 273, 282, 285, 286, 310, 318, 345, 357, 366, 370, 385, 399, 418, 430, 434, 442, 455, 465, 474, 483, 494, 498, 518, 555, 561, 574, 582, 595, 602, 609, 618, 642, 645, 651, 663, 665

Offset: 1

T. D. Noe, Apr 27 2011

nonn

The number pq+pr+qr is prime only if p, q, and r are distinct. The primes of form pq+pr+qr are in A087054. A prime may have multiple representations as pq+pr+qr; for example, 2*3*13 and 3*5*7 both produce the prime 71.

As mentioned by Ufnarovski and Ahlander, if pq+pr+qr is prime, then the arithmetic derivative (A003415) of pqr is that prime. They conjecture that this sequence and A087054 are infinite.

T. D. Noe, Table of n, a(n) for n = 1..1000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A003415, A087054, A356475.

pqr[nn_] := Module[{p=Prime[Range[PrimePi[nn/6]+1]],i,j,k,n,prod}, Sort[Reap[i=0; While[i++; p[[i]]p[[i+1]]p[[i+2]] <= nn, j=i; While[j++; p[[i]]p[[j]]p[[j+1]] <= nn, k=j; While[k++; prod=p[[i]]p[[j]]p[[k]]; prod <= nn, n=p[[i]]p[[j]]+p[[i]]p[[k]]+p[[j]]p[[k]]; If[PrimeQ[n], Sow[prod]]]]]][[2,1]]]]; pqr[1000]
Take[Union[Times@@@Select[Subsets[Prime[Range[30]],{3}],PrimeQ[ Total[ Times@@@Subsets[#,{2}]]]&]],60](* Harvey P. Dale, Dec 29 2011 *)

A356471 First of 5 consecutive primes p,q,r,s,t such that pq+ qr + rs + st + t*p is prime.

Original entry on oeis.org

19, 41, 47, 53, 157, 199, 491, 557, 563, 571, 647, 1063, 1091, 1097, 1109, 1163, 1171, 1217, 1259, 1279, 1361, 1367, 1487, 1601, 1621, 1753, 1901, 1951, 2053, 2161, 2383, 2441, 2549, 2777, 2851, 2879, 2887, 2953, 2957, 3041, 3061, 3067, 3163, 3191, 3491, 3499, 3719, 3881, 4003, 4007, 4013, 4093

Offset: 1

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

nonn

			a(3) = 47 is a term because 47, 53, 59, 61, 67 are 5 consecutive primes with 47*53 + 53*59 + 59*61 + 61*67 + 67*47 = 16453 prime.

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

Cf. A356475, A356477.

R:= NULL: count:= 0:
P:= Vector(5,ithprime):
while count < 100 do
  x:= P[1]*P[2]+P[2]*P[3]+P[3]*P[4]+P[4]*P[5]+P[5]*P[1];
  if isprime(x) then R:= R, P[1]; count:= count+1 fi;
  P[1..4]:= P[2..5];
  P[5]:= nextprime(P[5]);
od:
R;

Select[Partition[Prime[Range[600]], 5, 1], PrimeQ[Total[# * RotateLeft[#]]] &][[;; , 1]] (* Amiram Eldar, Aug 08 2022 *)

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q, r, s, t = 2, 3, 5, 7, 11
    while True:
        if isprime(p*q + q*r + r*s + s*t + t*p): yield p
        p, q, r, s, t = q, r, s, t, nextprime(t)
print(list(islice(agen(), 52))) # 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

cp's OEIS Frontend

A189759 Numbers pqr such that pq + pr + qr is prime, where p, q, and r are primes.

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A356471 First of 5 consecutive primes p,q,r,s,t such that pq+ qr + rs + st + t*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

A189759 Numbers pqr such that pq + pr + qr is prime, where p, q, and r are primes.

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Mathematica

A356471 First of 5 consecutive primes p,q,r,s,t such that p*q+ q*r + r*s + s*t + t*p is prime.

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Mathematica

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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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A356471 First of 5 consecutive primes p,q,r,s,t such that pq+ qr + rs + st + t*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.