A095100 Integers m of the form 4k+3 for which all sums Sum_{i=1..u} J(i/m) (with u ranging from 1 to (m-1)) are nonnegative, where J(i/m) is Jacobi symbol of i and m.

3, 7, 11, 15, 23, 27, 31, 35, 39, 47, 55, 59, 63, 71, 75, 79, 83, 87, 95, 103, 111, 119, 131, 135, 143, 151, 159, 167, 171, 175, 183, 191, 199, 215, 231, 239, 243, 251, 255, 263, 271, 279, 287, 295, 299, 303, 311, 319, 327, 335, 343, 351, 359, 363

Offset: 1

Author

Antti Karttunen, Jun 01 2004

Keywords

• nonn

Comments

Integers whose Jacobi-vector forms a valid Motzkin-path.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Crossrefs

Subset of A095102. Complement of A095101 in A004767.

Cf. A095090.

Programs

Mathematica

isMotzkin[n_, k_] := Module[{s = 0, r = True}, Do[s += JacobiSymbol[i, n]; If[s < 0, r = False; Break[]], {i, 1, k}]; r]; A095100[n_] := Select[4*Range[0, n+1]+3, isMotzkin[#, Quotient[#, 2]] &]; A095100[90] (* Jean-François Alcover, Oct 08 2013, translated from Sage *)

PARI

isok(m) = {if(m%4<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

Sage

def is_Motzkin(n, k):
    s = 0
    for i in range(1, k + 1) :
        s += jacobi_symbol(i, n)
        if s < 0: return False
    return True
def A095100_list(n):
    return [m for m in range(3, n + 1, 4) if is_Motzkin(m, m // 2)]
A095100_list(363) # Peter Luschny, Aug 08 2012

Formula

a(n) = 4*A095274(n) + 3.