OFFSET
1,1
COMMENTS
Note that the definition says "at least two", so a(n) = a(n) itself is not allowed as a possible sum (see Examples).
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
P. De Geest, WONplate 122
Sean A. Irvine, Java program (github)
C. Rivera, Puzzle 46
FORMULA
Prime(n) such that A307610(n) > 4. - Ray Chandler, Sep 21 2023
EXAMPLE
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
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Patrick De Geest, Feb 04 2002
EXTENSIONS
The terms have been confirmed by Sean A. Irvine, Dec 24 2021. - N. J. A. Sloane, Dec 25 2021
STATUS
approved