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A130621
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List the first term of each triple of consecutive primes with the property that their sum is the square of a prime.
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2
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13, 37, 277, 313, 613, 7591, 8209, 12157, 23053, 32233, 42953, 44887, 105649, 225769, 245941, 258707, 287671, 331333, 342049, 346111, 347443, 393853, 560719, 721267, 867253, 1001089, 1064431, 1219849, 1545127, 1556623, 1617727, 1752607
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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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(37, 41, 43) is a triple of consecutive prime numbers; their sum is 121 which is a prime squared. Hence 37 is in the sequence.
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MAPLE
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f:= proc(n) local p, q, r;
q:= prevprime(floor(n/3));
p:= prevprime(q);
r:= nextprime(q);
if p+q+r = n then return p
elif p+q+r < n then
while p+q+r < n do
p:= q; q:= r; r:= nextprime(r);
od;
if p+q+r = n then return p fi
else
while p+q+r > n do
r:= q; q:= p; p:= prevprime(p);
od;
if p+q+r = n then return p fi;
fi;
false
end proc:
R:= NULL: count:= 0:
p:= 3:
while count < 100 do
p:= nextprime(p);
v:= f(p^2);
if v::integer then
R:= R, v; count:= count+1;
fi
od:
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MATHEMATICA
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a={}; For[n=1, n<100000, n++, If[PrimeQ[Sqrt[Prime[n]+Prime[n+1]+Prime[n+2]]], AppendTo[a, Prime[n]]]]; a
Select[Partition[Prime[Range[132000]], 3, 1], PrimeQ[Sqrt[Total[#]]]&][[All, 1]] (* Harvey P. Dale, Dec 12 2022 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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