A109720 Periodic sequence {0,1,1,1,1,1,1} or n^6 mod 7.

0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1

Offset: 0

Author

Bruce Corrigan (scentman(AT)myfamily.com), Aug 09 2005

Keywords

• easy
• mult
• nonn

Comments

This sequence also represents n^12 mod 7; n^18 mod 7; (exponents are = 0 mod 6).

Characteristic sequence for numbers n>=1 to be relatively prime to 7. - Wolfdieter Lang, Oct 29 2008

a(n+4), n>=0, (periodic 1,1,1,0,1,1,1) is also the characteristic sequence for mod m reduced positive odd numbers (i.e., gcd(2*n+1,m)=1, n>=0) for each modulus m from 7*A003591 = [7,14,28,49,56,98,112,196,...]. [Wolfdieter Lang, Feb 04 2012]

Links

• Index entries for characteristic functions - _Reinhard Zumkeller_, Nov 30 2009
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1)

Crossrefs

Cf. A010876 = n mod 7; A053879 = n^2 mod 7; A070472 = m^3 mod 7; A070512 = n^4 mod 7; A070593 = n^5 mod 7.

Cf. A168185, A145568, A168184, A168182, A168181, A097325, A011558, A166486, A011655, A000035.

Programs

Mathematica

PadRight[{},120,{0,1,1,1,1,1,1}] (* Harvey P. Dale, Jul 09 2018 *)

PARI

a(n)=n^6%7 \\ Charles R Greathouse IV, Sep 24 2015

Sage

[power_mod(n,6,7)for n in range(0, 105)] # Zerinvary Lajos, Nov 06 2009

Formula

a(n) = 0 if n=0 mod 7; a(n)= 1 else.

G.f. = (x+x^2+x^3+x^4+x^5+x^6)/(1-x^7)= -x*(1+x)*(1+x+x^2)*(x^2-x+1) / ( (x-1)*(1+x+x^2+x^3+x^4+x^5+x^6) ).

a(n)=1-A082784(n); a(A047304(n))=1; a(A008589(n))=0; A033439(n) = SUM(a(k)*(n-k): 0<=k<=n). - Reinhard Zumkeller, Nov 30 2009

Multiplicative with a(p) = (if p=7 then 0 else 1), p prime. - Reinhard Zumkeller, Nov 30 2009

Dirichlet g.f. (1-7^(-s))*zeta(s). - R. J. Mathar, Mar 06 2011

For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013