A095102 - OEIS

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

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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