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

%I A084147 #17 Feb 16 2025 08:32:49
%S A084147 36,41,60,72,83,90,100,112,119,120,138,143,152,180,187,197,199,204,
%T A084147 210,221,223,228,251,258,276,281,300,304,323,330,372,384,390,395,401,
%U A084147 408,410,434,439,456,462,473,480,491,492,508,533,540,551,552,558,559,576
%N A084147 Integers that have exactly 2 representations as sums of consecutive primes.
%C A084147 More fundamental than A067372, which gives integers having 2 *or more* such representations
%H A084147 Robert Israel, <a href="/A084147/b084147.txt">Table of n, a(n) for n = 1..10000</a>
%H A084147 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%e A084147 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.
%p A084147 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
%p A084147 # Alternative
%p A084147 N:= 70: # for terms up to prime(N-1)+prime(N)
%p A084147 P:= [seq(ithprime(i),i=1..N)]: m:= P[N-1]+P[N]:
%p A084147 S:= ListTools:-PartialSums(P):
%p A084147 V:= Vector(m):
%p A084147 for i from 2 while S[i] <= m do V[S[i]]:= 1 od:
%p A084147 for i from 1 to N-2 do
%p A084147   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
%p A084147 od od:
%p A084147 select(t -> V[t] = 2, [$1..m]); # _Robert Israel_, Feb 14 2021
%t A084147 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 *)
%Y A084147 Cf. A067372, A084143.
%K A084147 nonn
%O A084147 1,1
%A A084147 _Eric W. Weisstein_, May 15 2003
%E A084147 More terms from _John W. Layman_, May 21 2003