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A067380 Primes expressible as the sum of (at least two) consecutive primes in at least 4 ways.

%I A067380 #36 Aug 11 2024 14:41:34
%S A067380 311,863,14369,14699,15329,19717,29033,34421,36467,37607,40433,42463,
%T A067380 48731,49253,49499,55813,67141,70429,76423,78791,85703,90011,94559,
%U A067380 97159,98411,109159,110359,110527,125821,130513,134921,141587,147031,147347,155087,155387
%N A067380 Primes expressible as the sum of (at least two) consecutive primes in at least 4 ways.
%C A067380 Note that the definition says "at least two", so a(n) = a(n) itself is not allowed as a possible sum (see Examples).
%H A067380 Jon E. Schoenfield, <a href="/A067380/b067380.txt">Table of n, a(n) for n = 1..10000</a>
%H A067380 P. De Geest, <a href="https://www.worldofnumbers.com/em122.htm">WONplate 122</a>
%H A067380 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a067/A067380.java">Java program</a> (github)
%H A067380 C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_046.htm">Puzzle 46</a>
%F A067380 Prime(n) such that A307610(n) > 4. - Ray Chandler, Sep 21 2023
%e A067380 311 is a term because 311 is prime and
%e A067380   11+13+17+19+23+29+31+37+41+43+47 = 311,
%e A067380   31+37+41+43+47+53+59 = 311,
%e A067380   53+59+61+67+71 = 311,
%e A067380   101+103+107 = 311.
%e A067380 1151 is not a term, since although 1151 is prime it only has three representations of the required form:
%e A067380   101+97+89+83+79+73+71+67+61+59+53+47+43+41+37+31+29+23+19+17+13+11+7 = 1151,
%e A067380   239+233+229+227+223 = 1151,
%e A067380   389+383+379 = 1151.
%e A067380 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
%t A067380 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 *)
%o A067380 (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
%Y A067380 Cf. A050936, A067372-A067381, A307610.
%K A067380 nonn
%O A067380 1,1
%A A067380 _Patrick De Geest_, Feb 04 2002
%E A067380 The terms have been confirmed by _Sean A. Irvine_, Dec 24 2021. - _N. J. A. Sloane_, Dec 25 2021