A112056 Odd numbers of the form 4n-1 for which Jacobi-first-non-one(4n-1) differs from Jacobi-first-non-one(4n+1).

47, 71, 119, 167, 191, 287, 311, 359, 407, 431, 479, 527, 551, 647, 671, 719, 767, 791, 839, 887, 911, 959, 1007, 1031, 1127, 1151, 1199, 1247, 1271, 1319, 1367, 1391, 1487, 1511, 1559, 1607, 1631, 1679, 1727, 1751, 1799, 1847, 1871, 1967

Offset: 1

Author

Antti Karttunen, Aug 27 2005

Keywords

• nonn

Comments

Here Jacobi-first-non-one(m) (for odd numbers m) is defined as the first value of i >= 1, for which Jacobi symbol J(i,m) is not +1 (i.e. is either 0 or -1).

Crossrefs

Cf. A112054, A112057, A112058.

Programs

Mathematica

a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; 4*Select[Range[1000], a112046[2#] - a112046[2# - 1] != 0 &] - 1 (* Indranil Ghosh, May 24 2017 *)

Python

from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([4*n - 1 for n in range(1, 1001) if a(n)!=0]) # Indranil Ghosh, May 24 2017

Formula

a(n) = 4*A112054(n)-1.

a(n) = A112057(n)-2 = A112058(n)-1.