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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).
12
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
Peter Borwein, Stephen K.K. Choi and Michael Coons, Completely multiplicative functions taking values in {-1,1}, arXiv:0809.1691 [math.NT], 2008.
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
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 01 2004
STATUS
approved