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A348353
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Primes p such that p^2+q+r, p+q^2+r and p+q+r^2 are all prime, where q and r are the next two primes after p.
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1
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11, 379, 419, 641, 7433, 10177, 32609, 33809, 37511, 41081, 54851, 58313, 63671, 68491, 71069, 85427, 106163, 112087, 116741, 117167, 127373, 128489, 137483, 155689, 157793, 180053, 197003, 208577, 209327, 210659, 213467, 227501, 234599, 248099, 276439, 278279, 288461, 298667, 313109, 317701, 325421
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OFFSET
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1,1
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COMMENTS
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The generalized Bunyakovsky conjecture implies that there are infinitely many terms, in fact infinitely many for which q = p+2 and r = p+6. - Robert Israel, Oct 21 2021
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LINKS
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EXAMPLE
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a(1) = 11 is a term because with p=11, q=13, r=17 we have p^2+q+r = 151, p+q^2+r = 197, p+q+r^2 = 313, all of which are primes.
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MAPLE
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R:= NULL: count:= 0:
p:= 2: q:= 3: r:= 5:
while count < 50 do
p:= q; q:= r; r:= nextprime(r);
if andmap(isprime, [p^2+q+r, p+q^2+r, p+q+r^2]) then
count:= count+1; R:= R, p;
fi
od:
R;
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MATHEMATICA
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Select[Partition[Select[Range[350000], PrimeQ], 3, 1], And @@ PrimeQ[{#[[1]]^2 + #[[2]] + #[[3]], #[[1]] + #[[2]]^2 + #[[3]], #[[1]] + #[[2]] + #[[3]]^2}] &][[;; , 1]] (* Amiram Eldar, Oct 14 2021 *)
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PROG
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(Python)
from itertools import islice
from sympy import nextprime, isprime
def A348353(): # generator of terms.
p, q, r = 2, 3, 5
while True:
if isprime(p*p+q+r) and isprime(p+q*q+r) and isprime(p+q+r*r):
yield p
p, q, r = q, r, nextprime(r)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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