A054998 Integers that can be expressed as the sum of consecutive primes in exactly 3 ways.

41, 83, 197, 199, 223, 240, 251, 281, 287, 340, 371, 401, 439, 491, 510, 593, 660, 733, 803, 857, 864, 883, 931, 941, 961, 983, 990, 991, 1012, 1060, 1061, 1099, 1104, 1187, 1236, 1283, 1313, 1361, 1381, 1392, 1433, 1439, 1493, 1511, 1523, 1524, 1553

Offset: 1

Author

Jud McCranie, May 30 2000

Keywords

• nonn

Examples

			41 can be expressed as 41 or 11+13+17 or 2+3+5+7+11+13, so 41 is in the sequence.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, section C2.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A054845, A054859, A054996, A054997, A054999, A055500, A055001.

Programs

Maple

N:= 10^4: # to get all terms <= N
P:= [0,op(select(isprime, [2,seq(i,i=3..N,2)]))]:
nP:= nops(P);
S:= ListTools:-PartialSums(P):
V:= Vector(N):
for i from 1 to nP-1 do
  for j from i+1 to nP while S[j] - S[i] <= N do
     V[S[j]-S[i]]:= V[S[j]-S[i]]+1
od od:
select(t -> V[t] = 3, [$1..N]): # Robert Israel, Apr 05 2017

Mathematica

Module[{nn = 300, s}, s = Array[Prime, nn]; Keys@ Take[Select[KeySort@ Merge[Table[PositionIndex@ Map[Total, Partition[s, k, 1]], {k, nn/2}], Identity], Length@ # == 3 &], Floor[nn/6]]] (* Michael De Vlieger, Apr 06 2017, Version 10 *)

Formula

A054845(a(n)) = 3. - Ray Chandler, Sep 20 2023