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

1,1

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime Sums

Example

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.

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 *)

Crossrefs

Cf. A067372, A084143.

Sequence in context: A295492 A225103 A067372 * A261258 A261372 A044862

Adjacent sequences: A084144 A084145 A084146 * A084148 A084149 A084150

Keyword

nonn

Author

Eric W. Weisstein, May 15 2003

Extensions

More terms from John W. Layman, May 21 2003

Status

approved