A225889 Least prime p_m such that n = p_m-p_{m-1}+...+(-1)^(m-k)*p_k for some 0<k<m, where p_j denotes the j-th prime.

3, 5, 7, 5, 7, 11, 13, 11, 11, 17, 19, 17, 17, 23, 17, 23, 23, 31, 23, 41, 23, 41, 31, 47, 29, 47, 37, 59, 41, 59, 37, 59, 43, 67, 37, 67, 43, 67, 43, 73, 61, 83, 53, 83, 47, 101, 61, 97, 53, 97, 59, 97, 59, 103, 61, 109, 67, 127, 67, 131

Offset

1,1

Comments

By a conjecture of the author, a(n) <= 2*n+2.2*sqrt(n), and moreover a(n) <= n+4.6*sqrt(n) if n is odd. Clearly a(n)>n. We guess that a(2n)/(2n) --> 2 as n tends to the infinity.

Note that this sequence is different from A222579 which involves a stronger conjecture of the author.

Zhi-Wei Sun also conjectured that any positive even integer m can be written in the form p_n-p_{n-1}+...+(-1)^{n-k}*p_k with k < n and 2m-3.6*sqrt(m+1) < p_n < 2m+2.2*sqrt(m).

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), no.8, 2794-2812.

Example

a(7) = 13 since 7 = 13-11+7-5+3.
a(20) = 41 since 20 = 41-37+31-29+23-19+17-13+11-7+5-3.

Mathematica

s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
Do[Do[If[s[j]-(-1)^(j-i)*s[i]==m, Print[m, " ", Prime[j]]; Goto[aa]], {j, PrimePi[m]+1, PrimePi[2m+2.2Sqrt[m]]}, {i, 0, j-2}];
Print[m, " ", counterexample]; Label[aa]; Continue, {m, 1, 100}]

Crossrefs

Cf. A000040, A222579.

Sequence in context: A141710 A279399 A321784 * A070647 A070949 A222579

Adjacent sequences: A225886 A225887 A225888 * A225890 A225891 A225892

Keyword

nonn

Author

Zhi-Wei Sun, May 19 2013

Status

approved