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

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

Offset: 1

Author

Patrick De Geest, Feb 04 2002

Keywords

• nonn

Comments

Note that the definition says "at least two", so a(n) = a(n) itself is not allowed as a possible sum (see Examples).

Examples

			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
		

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

Crossrefs

Cf. A050936, A067372-A067381, A307610.

Programs

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

Mathematica

m=7!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[PrimeQ[p]&&pVladimir Joseph Stephan Orlovsky, Aug 15 2009 *)

Formula

Prime(n) such that A307610(n) > 4. - Ray Chandler, Sep 21 2023

Extensions

The terms have been confirmed by Sean A. Irvine, Dec 24 2021. - N. J. A. Sloane, Dec 25 2021