A055095 a(n) = 2*A000120(A003188(A055094(n))) - (n-1) = 2*A005811(A055094(n)) - (n-1).

0, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 1, 2, 3, 2, -3, 2, 7, 2, -3, 4, 3, 2, -3, 14, 1, 10, -3, 2, 3, 2, -11, 4, 1, -2, -7, 2, 3, 2, -11, 2, 7, 2, -7, -4, 3, 2, -19, 8, 25, 2, -11, 2, 19, -6, -15, 4, 1, 2, -19, 2, 3, -6, -23, -10, 7, 2, -15, 4, -5, 2, -27, 2, 1, 6, -15, -4, 3, 2, -39, 28, 1, 2, -27, -14, 3, 2, -27, 2, -9, -10, -19, 4, 3, -14, -47, 2, 15, -14, -19, 2, 3, 2, -35, -24

Offset

1,3

Comments

For all odd primes p, a(p) = +2 because Sum_{a=1..(p-2)} L((a(a+1))/p) = Sum_{a=1..(p-2)} L((1+(a^-1))/p) = -1; i.e. in Gray code expansion of A055094[p], the number of 1-bits is number of 0-bits + 2. However, a(n) = +2 also for some nonprime odd n = A055131.

References

See problem 9.2.2 in Elementary Number Theory by David M. Burton, ISBN 0-205-06978-9

Links

Indranil Ghosh, Table of n, a(n) for n = 1..4096

Formula

a(n) = (2*wt(GrayCode(qrs2bincode(n))))-(n-1).

Maple

A055095 := proc(n)
    2*A005811(A055094(n))-n+1 ;
end proc:
seq(A055095(n), n=1..20) ; # R. J. Mathar, Mar 10 2015

Mathematica

A005811[n_] := Length[Length /@ Split[IntegerDigits[n, 2]]];
A055094[n_] := With[{rr = Table[Mod[k^2, n], {k, 1, n-1}] // Union}, Boole[ MemberQ[rr, #]]& /@ Range[n-1]] // FromDigits[#, 2]&;
a[1] = 0; a[n_] := 2*A005811[A055094[n]] - (n-1);
Array[a, 105] (* Jean-François Alcover, Mar 05 2016 *)

Prog

(Python)
from sympy.ntheory.residue_ntheory import quadratic_residues as q
def a055094(n):
    Q=q(n)
    z=0
    for i in range(1, n):
        z*=2
        if i in Q: z+=1
    return z
def a005811(n): return bin(n^(n>>1))[2:].count("1")
def a(n): return 0 if n == 1 else 2*a005811(a055094(n)) - (n - 1) # Indranil Ghosh, May 13 2017

Crossrefs

Sequence in context: A059982 A187801 A134388 * A366770 A355249 A337618

Adjacent sequences: A055092 A055093 A055094 * A055096 A055097 A055098

Keyword

sign

Author

Antti Karttunen, Apr 04 2000

Status

approved