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A055095 a(n) = 2*A000120(A003188(A055094(n))) - (n-1) = 2*A005811(A055094(n)) - (n-1). 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 xrange(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 * A048685 A101050 A128979

Adjacent sequences:  A055092 A055093 A055094 * A055096 A055097 A055098

KEYWORD

sign

AUTHOR

Antti Karttunen, Apr 04 2000

STATUS

approved

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Last modified December 13 19:20 EST 2017. Contains 295976 sequences.