A084147 Integers that have exactly 2 representations as sums of consecutive primes.

36, 41, 60, 72, 83, 90, 100, 112, 119, 120, 138, 143, 152, 180, 187, 197, 199, 204, 210, 221, 223, 228, 251, 258, 276, 281, 300, 304, 323, 330, 372, 384, 390, 395, 401, 408, 410, 434, 439, 456, 462, 473, 480, 491, 492, 508, 533, 540, 551, 552, 558, 559, 576

Offset: 1

Author

Eric W. Weisstein, May 15 2003

Keywords

• nonn

Comments

More fundamental than A067372, which gives integers having 2 *or more* such representations

Examples

			36 is in the sequence because it can be written in exactly two ways as sum of consecutive primes: 17+19 and 5+7+11+13.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Sums

Crossrefs

Cf. A067372, A084143.

Programs

Maple

g:=sum(sum(product(x^ithprime(k),k=i..j),j=i+1..150),i=1..150): gser:=series(g,x=0,605): a:=proc(n) if coeff(gser,x^n)=2 then op(2,x^n) else fi end: seq(a(n),n=1..600); # Emeric Deutsch, Mar 30 2006
# Alternative
N:= 70: # for terms up to prime(N-1)+prime(N)
P:= [seq(ithprime(i),i=1..N)]: m:= P[N-1]+P[N]:
S:= ListTools:-PartialSums(P):
V:= Vector(m):
for i from 2 while S[i] <= m do V[S[i]]:= 1 od:
for i from 1 to N-2 do
  for j from i+2 to N while S[j]-S[i] <= m do V[S[j]-S[i]]:= V[S[j]-S[i]] + 1
od od:
select(t -> V[t] = 2, [$1..m]); # Robert Israel, Feb 14 2021

Mathematica

With[{nn=100},Take[Sort[Select[Tally[Flatten[Table[Total/@Partition[Prime[Range[nn]],n,1],{n,2,nn}]]],#[[2]]==2&]][[All,1]],nn]] (* Harvey P. Dale, Mar 06 2020 *)

Extensions

More terms from John W. Layman, May 21 2003