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A340053
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First of three consecutive primes p,q,r such that p*q*r mod (p+q), p*q*r mod (p+r) and p*q*r mod (q+r) are all primes.
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1
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13, 17, 37, 41, 43, 73, 89, 103, 127, 151, 157, 167, 239, 257, 271, 281, 337, 353, 379, 401, 409, 419, 421, 443, 521, 557, 587, 601, 607, 683, 719, 743, 839, 883, 941, 953, 983, 1093, 1097, 1297, 1361, 1409, 1427, 1429, 1511, 1543, 1583, 1597, 1637, 1657, 1741, 1801, 1913, 1973, 1993, 2017, 2281
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OFFSET
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1,1
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LINKS
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EXAMPLE
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a(3) = 37 is a term because 37, 41, 43 are consecutive primes, 37*41*43 = 65231, and 65231 mod (37+41) = 23, 65231 mod (37+43) = 31, and 65321 mod (41+43) = 47 are all primes.
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MAPLE
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q:= 2: r:= 3:
R:= NULL: count:= 0:
while count < 100 do
p:= q; q:= r; r:= nextprime(r);
if isprime(p*q*r mod (p+q)) and isprime(p*q*r mod (p+r)) and isprime(p*q*r mod (q+r)) then
count:= count+1; R:= R, p;
fi
od:
R;
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MATHEMATICA
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tcpQ[{a_, b_, c_}]:=Module[{t=a b c}, AllTrue[{Mod[t, a+b], Mod[t, b+c], Mod[t, a+c]}, PrimeQ]]; Select[Partition[Prime[Range[400]], 3, 1], tcpQ][[;; , 1]] (* Harvey P. Dale, Jul 21 2023 *)
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PROG
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(PARI) isok(p) = if (isprime(p), my(q=nextprime(p+1), r = nextprime(q+1), pqr = p*q*r); isprime(pqr % (p+q)) && isprime(pqr % (p+r)) && isprime(pqr % (q+r))); \\ Michel Marcus, Dec 28 2020
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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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