A226115 Least positive integer not of the form p_m - p_{m-1} + ... +(-1)^(m-k)*p_k with 0 < k < m <= n, where p_j denotes the j-th prime.

1, 2, 3, 6, 7, 10, 11, 14, 18, 18, 20, 20, 24, 24, 28, 28, 34, 34, 40, 40, 42, 42, 46, 46, 46, 54, 56, 56, 58, 58, 60, 64, 78, 78, 80, 80, 94, 94, 98, 98, 104, 104, 106, 106, 106, 106, 118, 118, 118, 118, 122, 122, 140, 140, 146, 146, 152, 152, 158, 158

Offset: 1

Author

Zhi-Wei Sun, May 27 2013

Keywords

• nonn

Comments

Conjecture: sqrt(2*a(n)) > sqrt(p_n)-0.7 for all n > 0, and a(n) is even for any n > 7.

Note that f(n) = sqrt(2*a(n))-sqrt(p_n)+0.7 is approximately equal to 0.000864 at n = 651. It seems that f(n) > 0.1 for any other value of n.

Examples

			a(4) = 6,  since 2,3,5,7 are the initial four primes, and 1=3-2, 2=5-3, 3=7-5+3-2, 4=5-3+2, 5=7-5+3.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), 2794-2812.

Crossrefs

Cf. A000040, A225889, A222579, A222580.

Programs

Mathematica

s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
R[j_]:=R[j]=Union[Table[s[j]-(-1)^(j-i)*s[i],{i,0,j-2}]]
t=1
Do[Do[Do[If[MemberQ[R[j],m]==True,Goto[aa]],{j,PrimePi[m]+1,n}];Print[n," ",m];t=m;Goto[bb];
Label[aa];Continue,{m,t,Prime[n]-1}];Print[n," ",counterexample];Label[bb],{n,1,100}]