A080114 Odd primes p for which all sums Sum_{j=1..u} L(j/p) (with u ranging from 1 to (p-1)/2) are nonnegative, where L(j/p) is Legendre symbol of j and p.

3, 5, 7, 11, 13, 23, 31, 37, 47, 59, 71, 79, 83, 103, 131, 151, 167, 191, 199, 239, 251, 263, 271, 311, 359, 383, 419, 431, 439, 479, 503, 563, 599, 607, 647, 659, 719, 743, 751, 839, 863, 887, 911, 919, 971, 983, 991, 1031, 1039, 1063, 1091, 1103, 1151, 1223

Offset: 1

Author

Antti Karttunen, Feb 11 2003

Keywords

• nonn

Comments

This sequence contains those 4k+1 primes p for which the first half (the (p-1)/2 most significant bits) of A055094(p) is in A014486 and those 4k+3 primes q, for which the whole A055094(q) is in A014486.

Are the 2nd, 5th and 8th primes (5,13,37) only terms of this sequence that are of the form 4k+1? [Searched up to a(211)=7927 by AK.]

No other such terms up to 19997. - Michel Marcus, Sep 21 2022

Links

• Michel Marcus, Table of n, a(n) for n = 1..430
• Antti Karttunen, Illustration of Legendre's candelabras
• Eric Weisstein's World of Mathematics, Legendre Symbol
• Wikipedia, Legendre symbol

Crossrefs

Cf. A080112, A080115. These are the primes for which a "Legendre's candelabra" can be constructed, see A080120.

Supersequence of A095102.

Programs

Maple

with(numtheory); # For ithprime and legendre.
A080114 := n -> ithprime(A080112(n));
A080114v2 := proc(upto_n) local j,a,p,i,s; a := []; for i from 2 to upto_n do p := ithprime(i); s := 0; for j from 1 to (p-1)/2 do s := s + legendre(j,p); if(s < 0) then break; fi; od; if(s >= 0) then a := [op(a),p]; fi; od; RETURN(a); end;

Mathematica

s[p_, u_] := Sum[JacobiSymbol[j, p], {j, 1, u}]; Select[Prime[Range[2, 200] ], (p = #; AllTrue[Range[(p - 1)/2], s[p, #] >= 0 &]) &] (* Jean-François Alcover, Mar 04 2016 *)

PARI

isok(p) = if (isprime(p) && (p>2), for (u=1, (p-1)/2, if (sum(j=1, u, kronecker(j, p)) < 0, return(0));); return(1);); \\ Michel Marcus, Sep 20 2022

Sage

def A080114_list(n) :
    a = []
    for i in (2..n) :
        s = 0
        p = nth_prime(i)
        for j in (1..(p-1)/2) :
            s += legendre_symbol(j, p)
            if s < 0 : break
        if s >= 0 : a.append(p)
    return a
A080114_list(200) # Peter Luschny, Aug 08 2012