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%I #11 Dec 25 2021 20:40:04
%S 34421,229841,235493,271919,345011,358877,414221,442019,488603,532823,
%T 621937,655561,824099,888793,896341,935791,954623,963173,988321,
%U 1055969,1083371,1083941,1115911,1170857,1261763,1338823,1352863,1409299,1444957,1598953,1690597
%N Primes expressible as the sum of (at least two) consecutive primes in at least 5 ways.
%C Subsequence of A067380.
%H Jon E. Schoenfield, <a href="/A164556/b164556.txt">Table of n, a(n) for n = 1..3000</a>
%F A067375 INTERSECT A000040.
%e 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
%e 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)
%e are expressible in 5 ways as the sum of two or more consecutive primes.
%t 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={};
%t 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]
%o (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
%Y Cf. A067377, A067378, A067379, A067380, A067381.
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Aug 15 2009
%E Examples added by _R. J. Mathar_, Aug 19 2009
%E a(10)-a(28) from _Donovan Johnson_, Sep 16 2009
%E a(29)-a(31) from _Jon E. Schoenfield_, Dec 25 2021
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