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%I A050940 #29 Sep 20 2023 17:13:54
%S A050940 0,1,4,6,9,14,16,20,21,22,25,27,32,33,34,35,38,40,44,45,46,50,51,54,
%T A050940 55,57,62,63,64,65,66,69,70,74,76,80,81,82,85,86,87,91,92,93,94,96,99,
%U A050940 104,105,106,108,110,111,114,115,116,117,118,122,123,125
%N A050940 Numbers that are not the sum (of a nonempty sequence) of consecutive primes.
%C A050940 Where is there a proof that this sequence is infinite? - _Carlos Rivera_, Apr 17 2002
%C A050940 Moser shows that the average order of A054845 is log(2), and hence this sequence is infinite with lower density at least 1 - log 2 = 0.306.... - _Charles R Greathouse IV_, Mar 21 2011
%H A050940 Franklin T. Adams-Watters, <a href="/A050940/b050940.txt">Table of n, a(n) for n = 1..10001</a>
%H A050940 Leo Moser, <a href="https://doi.org/10.4153/CMB-1963-013-1">Notes on number theory. III. On the sum of consecutive primes</a>, Canad. Math. Bull. 6 (1963), pp. 159-161.
%F A050940 A054845(a(n)) = 0. - _Ray Chandler_, Sep 20 2023
%e A050940 The number 14 cannot be expressed as a sum of any consecutive subset of the following primes: {2, 3, 5, 7, 11, 13}.
%o A050940 (BASIC) 10 N=1 20 N=N+1: if N=prmdiv(N) then goto 20 30 P=1 40 P=nxtprm(P):S=P:Q=P: if S>N\2 then print N;:goto 20 50 Q=nxtprm(Q):S=S+Q 60 if S=N then goto 20 70 if S>N then goto 40 80 goto 50
%o A050940 (PARI) is(n)=if(isprime(n), return(0)); my(v,m=1,t); while(1, v=vector(m++); v[m\2]=precprime(n\m); for(i=m\2+1,m,v[i]=nextprime(v[i-1]+1)); forstep(i=m\2-1,1,-1,v[i]=precprime(v[i+1]-1)); if(v[1]==0, return(1)); t=vecsum(v); if (t==n, return(0)); if(t>n, while(t>n, t-=v[m]; v=concat(precprime(v[1]-1), v[1..m-1]); t+=v[1]), while(t<n, t-=v[1]; v=concat(v[2..m], nextprime(v[m]+1)); t+=v[m])); if(v[1]==0, return(1)); if(t==n, return(0))) \\ _Charles R Greathouse IV_, May 05 2016
%Y A050940 Complement of A034707.
%K A050940 nonn
%O A050940 1,3
%A A050940 _N. J. A. Sloane_, Jan 02 2000

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