A181901 a(n) = least positive integer m such that 2(s_k)^2 for k=1,...,n are pairwise distinct modulo m where s_k = Sum_{j=1..k} (-1)^(k-j)*p_j and p_j is the j-th prime.

1, 4, 7, 9, 13, 17, 19, 23, 25, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Author

Zhi-Wei Sun, Mar 31 2012

Keywords

• nonn

Comments

On Mar 28 2012, Zhi-Wei Sun conjectured that a(n) is the (n+1)-th prime p_{n+1} with the only exceptions being a(1)=1, a(2)=4, a(4)=9 and a(9)=25. He has shown that 2(s_k)^2 (k=1,...,n) are indeed pairwise distinct modulo p_{n+1} and hence a(n) does not exceed p_{n+1}.

Note that the sequence 0,s_1,s_2,s_3,... is A008347 introduced by N. J. A. Sloane and J. H. Conway.

Compare a(n) with the sequence A210640.

The conjecture was verified for n up to 2*10^5 by the author in 2018, and for n up to 3*10^5 by Chang Zhang (a student at Nanjing Univ.) in June 2020. - Zhi-Wei Sun, Jun 22 2020

Examples

			We have a(4)=9 since 2(s_1)^2=8, 2(s_2)^2=2, 2(s_3)^2=32, 2(s_4)^2=18 are pairwise distinct modulo 9 but not pairwise distinct modulo any of 1,...,8.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..600
• Zhi-Wei Sun, An amazing recurrence for primes, a message to Number Theory List, March 31, 2012.
• Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Crossrefs

Cf. A000040, A008347, A210640, A210393, A210394, A210186, A210144, A208494, A208643, A207982.

Programs

Mathematica

s[n_]:=Sum[(-1)^k*Prime[k],{k,1,n}]
f[n_]:=2*s[n]^2
R[n_,m_]:=Union[Table[Mod[f[k],m],{k,1,n}]]
Do[Do[If[Length[R[n,m]]==n,Print[n," ",m];Goto[aa]],{m,1,Prime[n+1]}];
   Print[n];Label[aa];Continue,{n,1,600}]