A095102 Odd primes p for which all sums Sum_{i=1..u} L(i/p) (with u ranging from 1 to (p-1)) are nonnegative, where L(i/p) is Legendre symbol of i and p, defined to be 1 if i is a quadratic residue (mod p) and -1 if i is a quadratic non-residue (mod p).

3, 7, 11, 23, 31, 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

Offset

1,1

Comments

All 4k+3 primes whose Legendre-vector (cf. A055094) forms a valid Dyck-path (cf. A014486).

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Peter Borwein, Stephen K.K. Choi and Michael Coons, Completely multiplicative functions taking values in {-1,1}, arXiv:0809.1691 [math.NT], 2008.

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Formula

a(n) = 4*A095272(n) + 3.

Mathematica

isMotzkin[n_, k_] := Module[{s = 0, r = True}, Do[s += JacobiSymbol[i, n]; If[s < 0, r = False; Break[]], {i, 1, k}]; r]; A095102[max_] := Select[ Range[3, max, 4], PrimeQ[#] && isMotzkin[#, Quotient[#, 2]]&]; A095102[1151] (* Jean-François Alcover, Feb 16 2018, after Peter Luschny *)

Prog

(Sage)
def A095102_list(n) :
    def is_Motzkin(n, k):
        s = 0
        for i in (1..k):
            s += jacobi_symbol(i, n)
            if s < 0: return False
        return True
    P = filter(is_prime, range(3, n+1, 4))
    return filter(lambda m: is_Motzkin(m, m//2), P)
A095102_list(1151) # Peter Luschny, Aug 09 2012
(PARI) isok(m) = {if(!isprime(m-(m<3)), return(0)); my(s=0); for(i=1, m-1, if((s+=kronecker(i, m))<0, return(0))); 1; } \\ Jinyuan Wang, Jul 20 2020

Crossrefs

Intersection of A000040 and A095100. Subset of A080114 (see comments there). Complement of A095103 in A002145.

Cf. A095092.

Sequence in context: A239227 A154427 A287459 * A192614 A112715 A106935

Adjacent sequences: A095099 A095100 A095101 * A095103 A095104 A095105

Keyword

nonn

Author

Antti Karttunen, Jun 01 2004

Status

approved