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A067380
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Primes expressible as the sum of (at least two) consecutive primes in at least 4 ways.
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5
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311, 863, 14369, 14699, 15329, 19717, 29033, 34421, 36467, 37607, 40433, 42463, 48731, 49253, 49499, 55813, 67141, 70429, 76423, 78791, 85703, 90011, 94559, 97159, 98411, 109159, 110359, 110527, 125821, 130513, 134921, 141587, 147031, 147347, 155087, 155387
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OFFSET
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1,1
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COMMENTS
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Note that the definition says "at least two", so a(n) = a(n) itself is not allowed as a possible sum (see Examples).
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LINKS
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FORMULA
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Prime(n) such that A307610(n) > 4. - Ray Chandler, Sep 21 2023
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EXAMPLE
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311 is a term because 311 is prime and
11+13+17+19+23+29+31+37+41+43+47 = 311,
31+37+41+43+47+53+59 = 311,
53+59+61+67+71 = 311,
101+103+107 = 311.
1151 is not a term, since although 1151 is prime it only has three representations of the required form:
101+97+89+83+79+73+71+67+61+59+53+47+43+41+37+31+29+23+19+17+13+11+7 = 1151,
239+233+229+227+223 = 1151,
389+383+379 = 1151.
Also, 16277 is not a term because although it has five representations as a sum of consecutive primes, it is not itself a prime. - Sean A. Irvine, Dec 25 2021
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MATHEMATICA
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m=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]], AppendTo[lst, lst1[[n]]]], {n, Length[lst1]-3}]; Union[lst] (* Vladimir Joseph Stephan Orlovsky, Aug 15 2009 *)
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PROG
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(Magma) M:=160000; 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 4]; // 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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