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A164556
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Primes expressible as the sum of (at least two) consecutive primes in at least 5 ways.
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3
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34421, 229841, 235493, 271919, 345011, 358877, 414221, 442019, 488603, 532823, 621937, 655561, 824099, 888793, 896341, 935791, 954623, 963173, 988321, 1055969, 1083371, 1083941, 1115911, 1170857, 1261763, 1338823, 1352863, 1409299, 1444957, 1598953, 1690597
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 34421 = Sum_{i=57..127} prime(i) = Sum_{i=226..248} prime(i) = Sum_{i=527..535} prime(i) = Sum_{i=654..660} prime(i) = Sum_{i=1382..1384} prime(i) and
a(3) = 235493 = Sum_{i=50..284} prime(i) = Sum_{i=120..300} prime(i) = Sum_{i=123..301} prime(i) = Sum_{i=334..424} prime(i) = Sum_{i=7701..7703} prime(i)
are expressible in 5 ways as the sum of two or more consecutive primes.
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MATHEMATICA
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m=3*7!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[PrimeQ[p]&&p<Prime[m]*3+8, AppendTo[lst, p]], {b, a+1, m, 1}], {a, m}]; lst1=Sort[lst]; lst={};
Do[If[lst1[[n]]==lst1[[n+1]]&&lst1[[n]]==lst1[[n+2]]&&lst1[[n]]==lst1[[n+3]]&&lst1[[n]]==lst1[[n+4]], AppendTo[lst, lst1[[n]]]], {n, Length[lst1]-4}]; Union[lst]
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PROG
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(Magma) M:=1695000; P:=PrimesUpTo(M); S:=[0]; for p in P do t:=S[#S]+p; if #S ge 3 then if t-S[#S-2] gt M then break; end if; end if; S[#S+1]:=t; end for; c:=[0:j in [1..M]]; for C in [2..#S-1] do if IsEven(C) then L:=1; else L:=#S-C; end if; for j in [1..L] do s:=S[j+C]-S[j]; if s gt M then break; end if; if IsPrime(s) then c[s]+:=1; end if; end for; end for; [j:j in [1..M]|c[j] ge 5]; // Jon E. Schoenfield, Dec 25 2021
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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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