A050940 Numbers that are not the sum (of a nonempty sequence) of consecutive primes.

0, 1, 4, 6, 9, 14, 16, 20, 21, 22, 25, 27, 32, 33, 34, 35, 38, 40, 44, 45, 46, 50, 51, 54, 55, 57, 62, 63, 64, 65, 66, 69, 70, 74, 76, 80, 81, 82, 85, 86, 87, 91, 92, 93, 94, 96, 99, 104, 105, 106, 108, 110, 111, 114, 115, 116, 117, 118, 122, 123, 125

Offset: 1

Author

N. J. A. Sloane, Jan 02 2000

Keywords

• nonn

Comments

Where is there a proof that this sequence is infinite? - Carlos Rivera, Apr 17 2002

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

Examples

			The number 14 cannot be expressed as a sum of any consecutive subset of the following primes: {2, 3, 5, 7, 11, 13}.
		

Links

• Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001
• Leo Moser, Notes on number theory. III. On the sum of consecutive primes, Canad. Math. Bull. 6 (1963), pp. 159-161.

Crossrefs

Complement of A034707.

Programs

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

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(tCharles R Greathouse IV, May 05 2016

Formula

A054845(a(n)) = 0. - Ray Chandler, Sep 20 2023