A010875 -id:A010875 - cp's OEIS frontend

A010886 Period 7: repeat [1, 2, 3, 4, 5, 6, 7].

1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4

Offset: 0

N. J. A. Sloane

Partial sums are given by A130485(n)+n+1. - Hieronymus Fischer, Jun 08 2007

Decimal expansion of 1234567/9999999 = 0.123456712345671234567... - Eric Desbiaux, Nov 03 2008

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1).

Cf. A002264, A002265, A002266, A004526, A010872, A010873, A010874, A010875, A010876, A010877, A130485.

Cf. A177160 (decimal expansion of (4502+sqrt(29964677))/6961).

&cat [[1, 2, 3, 4, 5, 6, 7]^^20]; // Wesley Ivan Hurt, Jul 18 2016

seq(op([1, 2, 3, 4, 5, 6, 7]), n=0..20); # Wesley Ivan Hurt, Jul 18 2016

PadRight[{}, 100, {1, 2, 3, 4, 5, 6, 7}] (* Wesley Ivan Hurt, Jul 18 2016 *)

a(n)=n%7+1 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = 1 + (n mod 7). - Paolo P. Lava, Nov 21 2006
a(n) = A010876(n) + 1. G.f.: (Sum_{k=0..6} (k+1)*x^k)/(1-x^7). Also (7*x^8-8*x^7+1)/((1-x^7)*(1-x)^2). - Hieronymus Fischer, Jun 08 2007
From Wesley Ivan Hurt, Jul 18 2016: (Start)
a(n) = a(n-7) for n>6.
a(n) = 1 - 6*floor(n/7) + Sum_{k=1..6} floor((n + k)/7). (End)

A038122 Start with {1,2,...,n}, replace any two numbers a,b with |a^2-b^2|, repeat until single number k remains; a(n) = minimal value of k.

Original entry on oeis.org

1, 3, 0, 16, 15, 63, 8, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0

Offset: 1

R. K. Guy

Due mostly to the efforts of Dean Hickerson, supported by David W. Wilson and Michael Kleber, it is now known that this has period 12 beginning at n=8.

			a(2) = 3 from (1,2); a(3) = 0 from ((1,2),3); a(4) = 16 from (((1,2),3),4); a(5) = 15 from ((((2,3),5),1),4)
a(6) = 63 from (((1,4),(3,5)),(2,6)) [ _Michael Kleber_ ]
a(7) = 8 from (((((4,5),6),(2,7)),1),3) [ Kleber ]
a(8) = 0 from ((((4,5),7)(2,6))((1,3),8)) [ Guy ]
a(9) = 3 from (2,(1,(((6,7),((3,4),8)),(5,9)))) [ Kleber ]
a(10)= 1 from ((((((((4,5),9),6),(8,10)),2),3),7),1) [ This and the following are due to _Dean Hickerson_ ]
a(11)= 0 from ((((((3,7),(9,11)),6),(8,10)),(1,2)),(4,5))
a(12)= 0 from ((((((1,3),7),(8,10)),(((5,6),9),(11,12))),2),4)
a(13)= 1 from (((((((((3,7),(9,11)),6),(8,10)),5),(12,13)),2),4),1) ...

Dean Hickerson and Michael Kleber, Reducing a Set by Subtracting Squares, J. Integer Sequences, Vol. 2, 1999, #4.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,-1,0,0,1).

LinearRecurrence[{0,0,1,0,0,-1,0,0,1},{1,3,0,16,15,63,8,0,3,1,0,0,1,3,0,4},120] (* Harvey P. Dale, Jul 29 2015 *)

a(n)=if(n<4||n>7, n*(n+1)/2%6, [16, 15, 63, 8][n-3]) \\ Charles R Greathouse IV, Feb 10 2017

def A038122(n): return (16,15,63,8)[n-4] if 3>1)%6 # Chai Wah Wu, Apr 17 2025

For n<4 and n>7, a(n) = n*(n+1)/2 mod 6 = A010875(A000217(n)). - Henry Bottomley, Feb 24 2003
a(n) = a(n-3)-a(n-6)+a(n-9) for n>16. - Colin Barker, Oct 01 2014
G.f.: x*(4*x^15 +60*x^14 +12*x^13 +8*x^12 -60*x^11 -12*x^10 -8*x^9 +60*x^8 +12*x^7 +7*x^6 -63*x^5 -12*x^4 -15*x^3 -3*x -1) / ((x -1)*(x^2 +1)*(x^2 +x +1)*(x^4 -x^2 +1)). - Colin Barker, Oct 01 2014

A070510 a(n) = n^3 mod 48.

Original entry on oeis.org

0, 1, 8, 27, 16, 29, 24, 7, 32, 9, 40, 35, 0, 37, 8, 15, 16, 17, 24, 43, 32, 45, 40, 23, 0, 25, 8, 3, 16, 5, 24, 31, 32, 33, 40, 11, 0, 13, 8, 39, 16, 41, 24, 19, 32, 21, 40, 47, 0, 1, 8, 27, 16, 29, 24, 7, 32, 9, 40, 35, 0, 37, 8, 15, 16, 17, 24, 43, 32, 45, 40, 23, 0, 25, 8, 3, 16, 5

Offset: 0

N. J. A. Sloane, May 12 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

[Modexp(n, 3, 48): n in [0..80]]; // Bruno Berselli, Mar 31 2016

PowerMod[Range[0,80],3,48] (* Harvey P. Dale, Apr 03 2013 *)

a(n)=n^3%48 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,3,48) for n in range(0, 78)] # Zerinvary Lajos, Oct 30 2009

From G. C. Greubel, Apr 01 2016: (Start)
a(n) = a(n-48).
a(12*m) = 0.
a(2*n) = 8*A010875(n). (End)

A104686 a(n) = n*(n+1)/2 (mod 6).

Original entry on oeis.org

0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3

Offset: 0

Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 22 2005

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,-1,0,0,1)

Table[Mod[(n^2 + n)/2, 6], {n, 0, 100}] (* Arkadiusz Wesolowski, Nov 28 2011 *)
Mod[Accumulate[Range[0,90]],6] (* or *) LinearRecurrence[{0,0,1,0,0,-1,0,0,1},{0,1,3,0,4,3,3,4,0},90] (* Harvey P. Dale, Aug 21 2022 *)

a(n) = A010875(A000217(n)). - Arkadiusz Wesolowski, Nov 28 2011

G.f.: -x*(1+3*x+3*x^3+3*x^5+x^6) / ( (x-1)*(1+x+x^2)*(x^2+1)*(x^4-x^2+1) ). - R. J. Mathar, Oct 16 2015

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

A126046 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 6.

Original entry on oeis.org

2, 3, 5, 1, 1, 5, 1, 1, 1, 5, 5, 1, 5, 1, 1, 1, 1, 1, 5, 1, 5, 5, 5, 5, 5, 1, 1, 5, 1, 1, 1, 5, 5, 1, 5, 5, 5, 5, 1, 1, 1, 5, 1, 5, 5, 1, 5, 5

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A010875, A126043-A126059.

Array[Mod[MersennePrimeExponent@ #, 6] &, 45] (* Michael De Vlieger, Apr 07 2018 *)

a(n) = A010875(A000043(n)). - Michel Marcus, Apr 07 2018

a(45)-a(46) from Ivan Panchenko, Apr 07 2018
a(47) from Ivan Panchenko, Apr 09 2018
a(48) from Max Alekseyev, Sep 19 2023

cp's OEIS Frontend

A010886 Period 7: repeat [1, 2, 3, 4, 5, 6, 7].

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A038122 Start with {1,2,...,n}, replace any two numbers a,b with |a^2-b^2|, repeat until single number k remains; a(n) = minimal value of k.

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A070510 a(n) = n^3 mod 48.

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A104686 a(n) = n*(n+1)/2 (mod 6).

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A282779 Period of cubes mod n.

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A126046 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 6.

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