A070472 -id:A070472 - cp's OEIS frontend

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

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

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]

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)

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.

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

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

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

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

A282779 Period of cubes mod n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 16, 17, 6, 19, 20, 21, 22, 23, 24, 25, 26, 9, 28, 29, 30, 31, 32, 33, 34, 35, 12, 37, 38, 39, 40, 41, 42, 43, 44, 15, 46, 47, 48, 49, 50, 51, 52, 53, 18, 55, 56, 57, 58, 59, 60, 61, 62, 21, 64, 65, 66, 67, 68, 69, 70, 71, 24, 73, 74, 75, 76, 77, 78, 79, 80, 27

Offset: 1

Ilya Gutkovskiy, Feb 21 2017

The length of the period of A000035 (n=2), A010872 (n=3), A109718 (n=4), A070471 (n=5), A010875 (n=6), A070472 (n=7), A109753 (n=8), A167176 (n=9), A008960 (n = 10), etc. (see also comment in A000578 from R. J. Mathar).
Conjecture: let a_p(n) be the length of the period of the sequence k^p mod n where p is a prime, then a_p(n) = n/p if n == 0 (mod p^2) else a_p(n) = n.
For example: sequence k^7 mod 98 gives 1, 30, 31, 18, 19, 48, 49, 50, 79, 80, 67, 68, 97, 0, 1, 30, 31, 18, 19, 48, 49, 50, 79, 80, 67, 68, 97, 0, ... (period 14), 7 is a prime, 98 == 0 (mod 7^2) and 98/7 = 14.

			a(9) = 3 because reading 1, 8, 27, 64, 125, 216, 343, 512, ... modulo 9 gives 1, 8, 0, 1, 8, 0, 1, 8, 0, ... with period length 3.

Ray Chandler, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Extended graphical example
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A000035, A000578, A008960, A010872, A010875, A046530, A070471, A070472, A109718, A109753, A167176, A186646.

a[1] = 1; a[n_] := For[k = 1, True, k++, If[Mod[k^3, n] == 0 && Mod[(k + 1)^3 , n] == 1, Return[k]]]; Table[a[n], {n, 1, 81}]

Apparently: a(n) = 2*a(n-9) - a(n-18).

Empirical g.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 + 3*x^8 + 8*x^9 + 7*x^10 + 6*x^11 + 5*x^12 + 4*x^13 + 3*x^14 + 2*x^15 + x^16) / ((1 - x)^2*(1 + x + x^2)^2*(1 + x^3 + x^6)^2). - Colin Barker, Feb 21 2017

A122606 n^(n+1) mod 7.

Original entry on oeis.org

0, 1, 1, 4, 2, 1, 6, 0, 1, 2, 5, 1, 5, 1, 0, 1, 4, 1, 4, 4, 6, 0, 1, 1, 3, 2, 6, 1, 0, 1, 2, 2, 1, 2, 6, 0, 1, 4, 6, 4, 3, 1, 0, 1, 1, 4, 2, 1, 6, 0, 1, 2, 5, 1, 5, 1, 0, 1, 4, 1, 4, 4, 6, 0, 1, 1, 3, 2, 6, 1, 0, 1, 2, 2, 1, 2, 6, 0, 1, 4, 6, 4, 3, 1, 0, 1, 1, 4, 2, 1, 6, 0, 1, 2, 5, 1, 5, 1, 0, 1, 4, 1, 4, 4, 6

Offset: 0

Zak Seidov, Sep 24 2006, Sep 25 2006

nonn

Cf. A053879, A070472, A007778.

Table[PowerMod[n,n+1,7],{n,0,120}] (* Harvey P. Dale, May 24 2012 *)

a(n) = n^(n+1) mod 7 = A007778(n) mod 7.

Periodic with cycle consisting of first 42 terms: {0,1,1,4,2,1,6,0,1,2,5,1,5,1,0,1,4,1,4,4,6,0,1,1,3,2,6,1,0,1,2,2,1,2,6,0,1,4,6,4,3,1}.

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

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A282779 Period of cubes mod n.

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A122606 n^(n+1) mod 7.

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