A070403 -id:A070403 - cp's OEIS frontend

A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].

1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5

Offset: 0

Paul Curtz, Dec 19 2008

Digital root of 2^n.
A regular version of Pitoun's sequence: a(n) = A029898(n+1).
Also obtained from permutations of A141425, A020806, A070366, A153110, A153990, A154127, A154687, or A154815.
This sequence and its (again period 6) repeated differences produce the table:
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4, -1, -2, -4,  1,  2,  4, -1, -2, ...
    1,  2, -5, -1, -2,  5,  1,  2, -5, -1, -2, ...
    1, -7,  4, -1,  7, -4,  1, -7,  4, -1,  7, ...
   -8, 11, -5,  8,-11,  5, -8, 11, -5,  8,-11, ...
   19,-16, 13,-19, 16,-13, 19,-16, 13,-19, 16, ...
  -35, 29,-32, 35,-29, 32,-35, 29,-32, 35,-29, ...
   64,-61, 67,-64, 61,-67, 64,-61, 67,-64, 61, ...
If each entry of this table is read modulo 9 we obtain the very regular table:
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
Also the decimal expansion of the constant 125/1001. - R. J. Mathar, Jan 23 2009
Digital root of the powers of any number congruent to 2 mod 9. - Alonso del Arte, Jan 26 2014

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

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

Cf. A004000, A007953, A010734, A010888, A029898, A030132, A082365, A145389, A189510.
Cf. digital roots of powers of c mod 9: c = 4, A100402; c = 5, A070366; c = 7, A070403; c = 8, A010689.
Cf. A020806, A070366, A141425, A153110, A153990, A154127, A154687, A154815.

&cat [[1, 2, 4, 8, 7, 5]^^30]; // Wesley Ivan Hurt, Jul 05 2016

seq(op([1, 2, 4, 8, 7, 5]), n=0..40); # Wesley Ivan Hurt, Jul 05 2016

Flatten[Table[{1, 2, 4, 8, 7, 5}, {20}]] (* Paul Curtz, Dec 19 2008 *)
Table[Mod[2^n, 9], {n, 0, 99}] (* Alonso del Arte, Jan 26 2014 *)

a(n)=lift(Mod(2,9)^n) \\ Charles R Greathouse IV, Apr 21 2015

a(n) + a(n+3) = 9 = A010734(n).
G.f.: (1+x+2x^2+5x^3)/((1-x)(1+x)(1-x+x^2)). - R. J. Mathar, Jan 23 2009
a(n) = A082365(n) mod 9. - Paul Curtz, Mar 31 2009
a(n) = -1/2*cos(Pi*n) - 3*cos(1/3*Pi*n) - 3^(1/2)*sin(1/3*Pi*n) + 9/2. - Leonid Bedratyuk, May 13 2012
a(n) = A010888(A004000(n+1)). - Ivan N. Ianakiev, Nov 27 2014
From Wesley Ivan Hurt, Apr 20 2015: (Start)
a(n) = a(n-6) for n>5.
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
a(n) = (2+3*(n-1 mod 3))*(n mod 2) + (1+3*(-n mod 3))*(n-1 mod 2). (End)
a(n) = 2^n mod 9. - Nikita Sadkov, Oct 06 2018
From Stefano Spezia, Mar 20 2025: (Start)
E.g.f.: 4*cosh(x) - exp(x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 5*sinh(x).
a(n) = A007953(2*a(n-1)) = A010888(2*a(n-1)). (End)

Edited by R. J. Mathar, Apr 09 2009

A010689 Periodic sequence: Repeat 1, 8.

Original entry on oeis.org

1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1

Offset: 0

N. J. A. Sloane

Also the digital root of 8^n. Also the decimal expansion of 2/11 = 0.181818181818... - Cino Hilliard, Dec 31 2004
Interleaving of A000012 and A010731. - Klaus Brockhaus, Apr 02 2010
Continued fraction expansion of (2 + sqrt(6))/4. - Klaus Brockhaus, Apr 02 2010
Digital root of the powers of any number congruent to 8 mod 9. - Alonso del Arte, Jan 26 2014

			0.18181818181818181818181818181818181818181...

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

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

Cf. A000012 (all 1's sequence), A010731 (all 8's sequence), A174925 (decimal expansion of (2 + sqrt(6))/4). [Klaus Brockhaus, Apr 02 2010]
Cf. Digital roots of powers of c mod 9: c = 2, A153130; c = 4, A100402; c = 5, A070366; c = 7, A070403.
Cf. sequences listed in Comments section of A283393.
Cf. A010888.

&cat[ [1, 8]: n in [0..52] ]; // Klaus Brockhaus, Apr 02 2010

&cat [[1,8]^^60]; // Bruno Berselli, Mar 10 2017

Table[Mod[8^n, 9], {n, 0, 99}] (* Alonso del Arte, Jan 26 2014 *)
PadRight[{},120,{1,8}] (* Harvey P. Dale, Jun 03 2015 *)

A010689(n):=if evenp(n) then 1 else 8$
makelist(A010689(n),n,0,30); /* Martin Ettl, Nov 09 2012 */

a(n)=1; if(n%2==1, 8, 1) \\ Felix Fröhlich, Aug 11 2014

[power_mod(8,n,9)for n in range(0,105)] # Zerinvary Lajos, Nov 27 2009

From Paul Barry, Sep 16 2004: (Start)
G.f.: (1 + 8*x)/((1 - x)*(1 + x)).
a(n) = (9 - 7*(-1)^n)/2.
a(n) = 8^(ceiling(n/2) - floor(n/2)).
a(n) = gcd((n-1)^3, (n+1)^3). (End)
E.g.f.: cosh(x) + 8*sinh(x). - Stefano Spezia, Feb 09 2025
a(n) = A010888(8*a(n-1)). - Stefano Spezia, Mar 20 2025

Definition edited and keywords cons, cofr added by Klaus Brockhaus, Apr 02 2010

A070366 a(n) = 5^n mod 9.

Original entry on oeis.org

1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7

Offset: 0

N. J. A. Sloane, May 12 2002

Period 6: repeat [1, 5, 7, 8, 4, 2].
Also the digital root of 5^n. - Cino Hilliard, Dec 31 2004
Digital root of the powers of any number congruent to 5 mod 9. - Alonso del Arte, Jan 26 2014

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

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

Cf. Digital roots of powers of c mod 9: c = 2, A153130; c = 4, A100402; c = 7, A070403; c = 8, A010689.

Cf. A000351, A007953, A010878, A010888.

[Modexp(5, n, 9): n in [0..100]]; // Wesley Ivan Hurt, Jun 28 2016

A070366:=n->power(5,n) mod 9: seq(A070366(n), n=0..100); # Wesley Ivan Hurt, Jun 28 2016

PowerMod[5, Range[0, 120], 9] (* Harvey P. Dale, Mar 27 2011 *)
Table[Mod[5^n, 9], {n, 0, 100}] (* G. C. Greubel, Mar 05 2016 *)

a(n)=lift(Mod(5,9)^n); \\ Charles R Greathouse IV, Sep 24 2015

From R. J. Mathar, Apr 20 2010: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
G.f.: ( 1+4*x+2*x^2+2*x^3 ) / ( (1-x)*(1+x)*(x^2-x+1) ). (End)
a(n) = 1/2^n (mod 9), n >= 0. - Wolfdieter Lang, Feb 18 2014
a(n) = A010878(A000351(n)). - Michel Marcus, Feb 20 2014
From G. C. Greubel, Mar 05 2016: (Start)
a(n) = a(n-6) for n>5.
E.g.f.: (1/2)*(9*exp(x) - exp(-x) + 2*sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2) - 6*exp(x/2)*cos(sqrt(3)*x/2)). (End)
a(n) = (9 - cos(n*Pi) - 6*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3))/2. - Wesley Ivan Hurt, Jun 28 2016
a(n) = 2^((-n) mod 6) mod 9. - Joe Slater, Mar 23 2017
a(n) = A007953(5*a(n-1)) = A010888(5*a(n-1)). - Stefano Spezia, Mar 20 2025

A100402 Digital root of 4^n.

Original entry on oeis.org

1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7

Offset: 0

Cino Hilliard, Dec 31 2004

Equals A141725 mod 9. - Paul Curtz, Sep 15 2008
Sequence is the digital root of A016777. - Odimar Fabeny, Sep 13 2010
Digital root of the powers of any number congruent to 4 mod 9. - Alonso del Arte, Jan 26 2014
Period 3: repeat [1, 4, 7]. - Wesley Ivan Hurt, Aug 26 2014
From Timothy L. Tiffin, Dec 02 2023: (Start)
The period 3 digits of this sequence are the same as those of A070403 (digital root of 7^n) but the order is different: [1, 4, 7] vs. [1, 7, 4].
The digits in this sequence appear in the decimal expansions of the following rational numbers: 49/333, 490/333, 4900/333, .... (End)

			4^2 = 16, digitalroot(16) = 7, the third entry.

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

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

Cf. Digital roots of powers of c mod 9: c = 2, A153130; c = 5, A070366; c = 7, A070403; c = 8, A010689.

Cf. A000302, A001022, A009966, A009975, A009984, A010872, A010888, A070403, A087752, A121013, A141725, A016777.

Table[PowerMod[4, n, 9], {n, 0, 100}] (* Timothy L. Tiffin, Dec 03 2023 *)
StringRepeat["1, 4, 7, ", 100] (* Timothy L. Tiffin, Dec 03 2023 *)

a(n)=[1,4,7][1+n%3]; \\ Joerg Arndt, Aug 26 2014

[power_mod(4, n, 9) for n in range(0, 105)] # Zerinvary Lajos, Nov 25 2009

a(n) = 4^n mod 9. - Zerinvary Lajos, Nov 25 2009
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-3) for n>2.
G.f.: (1+4*x+7*x^2)/ ((1-x)*(1+x+x^2)). (End)
a(n) = A010888(A000302(n)). - Michel Marcus, Aug 25 2014
a(n) = 3*A010872(n) + 1. - Robert Israel, Aug 25 2014
a(n) = 4 - 3*cos(2*n*Pi/3) - sqrt(3)*sin(2*n*Pi/3). - Wesley Ivan Hurt, Jun 30 2016
a(n) = A153130(2n). - Timothy L. Tiffin, Dec 01 2023
a(n) = A010888(A001022(n)) = A010888(A009966(n)) = A010888(A009975(n)) = A010888(A009984(n)) = A010888(A087752(n)) = A010888(A121013(n)). - Timothy L. Tiffin, Dec 02 2023
a(n) = A010888(4*a(n-1)). - Stefano Spezia, Mar 20 2025

A156760 5*4^n-1.

Original entry on oeis.org

4, 19, 79, 319, 1279, 5119, 20479, 81919, 327679, 1310719, 5242879, 20971519, 83886079, 335544319, 1342177279, 5368709119, 21474836479, 85899345919, 343597383679, 1374389534719, 5497558138879, 21990232555519, 87960930222079, 351843720888319

Offset: 0

Paul Curtz, Feb 15 2009

Second column of the array A132207, or, if this array is flattened, a(n)=A132207(A007583(n)).

			Binary.......................................Decimal
100................................................4
10011.............................................19
1001111...........................................79
100111111........................................319
10011111111.....................................1279
1001111111111...................................5119
100111111111111................................20479
10011111111111111..............................81919
1001111111111111111...........................327679
100111111111111111111........................1310719
10011111111111111111111......................5242879
1001111111111111111111111...................20971519
100111111111111111111111111.................83886079
10011111111111111111111111111..............335544319
1001111111111111111111111111111...........1342177279
... - _Philippe Deléham_, Feb 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500

[5*4^n-1: n in [0..30]]; // Vincenzo Librandi, Jul 02 2011

Table[5*4^n - 1, {n, 0, 18}] (* Michael De Vlieger, Apr 20 2015 *)

a(n) mod 9 = A070403(n+2).
a(n+1) = 10*A083420(n)+9 .
a(n) = 5*A000302(n)-1.
a(n) = ( A024036(n+1)+A140529(n) )/2.
a(n) = 4a(n-1)+3, a(0)=4.
a(n) = A003947(n+1)-1 = 5*a(n-1)-4*a(n-2). G.f.: (4-x)/((1-x)(1-4x)). - R. J. Mathar, Feb 23 2009
a(n) = A198693(n) + 2^(2n+1). - Bob Selcoe, Apr 20 2015

Edited and extended by R. J. Mathar, Feb 23 2009

A381487 Numbers which are a power of their digital root.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 81, 128, 256, 512, 729, 2401, 6561, 8192, 16384, 32768, 59049, 78125, 524288, 531441, 823543, 1048576, 2097152, 4782969, 33554432, 43046721, 67108864, 134217728, 282475249, 387420489, 1220703125, 2147483648, 3486784401, 4294967296

Offset: 1

Stefano Spezia, Feb 25 2025

			a(12) = 128 is a term since 128 = 2^7 = A010888(128)^7.

Michel Marcus, Table of n, a(n) for n = 1..3347

Cf. A010888, A023106, A381491, A381492.

Digital root of k^n: A000012 (1), A153130 (2), A100401 (3), A100402 (4), A070366 (5), A100403 (6), A070403 (7), A010689 (8), A010734 (9).

A010888[n_]:=n - 9*Floor[(n-1)/9]; kmax=5*10^6; a={0,1}; For[k=2, k<=kmax, k++, If[A010888[k]!=1, If[IntegerQ[Log[A010888[k],k]], AppendTo[a,k]]]]; a

isok(k) = if ((k==0) || (k==1), return(1)); my(d=(k-1)%9+1); if (d>1, d^logint(k, d) == k); \\ Michel Marcus, Feb 26 2025

lista(nn) = my(list = List()); listput(list, 0); listput(list, 1); for (n=2, 9, for (k=1, logint(nn, n), if ((n^k-1)%9+1 == n, listput(list, n^k)););); vecsort(Vec(list)); \\ Michel Marcus, Feb 27 2025

a(n) = A381491(n)^A381492(n).

A144449 a(n) = 4(4 + 9n^2 + 15*n).

Original entry on oeis.org

16, 112, 280, 520, 832, 1216, 1672, 2200, 2800, 3472, 4216, 5032, 5920, 6880, 7912, 9016, 10192, 11440, 12760, 14152, 15616, 17152, 18760, 20440, 22192, 24016, 25912, 27880, 29920, 32032, 34216, 36472, 38800, 41200, 43672, 46216, 48832, 51520, 54280, 57112

Offset: 0

Paul Curtz, Oct 06 2008

A decimation: A061039(6n+5).

a(n) mod 9 = period 3: repeat 7,4,1 = A070403(n+1).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A061039, A070403.

Subsequence of A008590.

[36*n^2 + 60*n + 16: n in [0..40]]; // Vincenzo Librandi, Aug 07 2011

Table[36n^2+60n+16,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{16,112,280},40] (* Harvey P. Dale, Apr 04 2020 *)

a(n)=36*n^2+60*n+16 \\ Charles R Greathouse IV, Jun 17 2017

[(6*n+5)^2 - 9 for n in (0..40)] # G. C. Greubel, Mar 06 2022

a(n) = a(n-1) + 24*(3*n+1) = a(n-1) + 72*n + 24, a(0)=16.
A061039(6n+2) = A061039(6n-4) + 24*(3n+1) = a(6n-4) + 72*n + 24, a(2)=16.
From G. C. Greubel, Mar 06 2022: (Start)
G.f.: 8*(2 + 8*x - x^2)/(1-x)^3.
E.g.f.: 4*(4 + 24*x + 9*x^2)*exp(x). (End)
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 1/12.
Sum_{n>=0} (-1)^n/a(n) = Pi/(18*sqrt(3)) + log(2)/18 - 1/12. (End)

Edited by Charles R Greathouse IV, Jul 25 2010

A070517 a(n) = n^4 mod 13.

Original entry on oeis.org

0, 1, 3, 3, 9, 1, 9, 9, 1, 9, 3, 3, 1, 0, 1, 3, 3, 9, 1, 9, 9, 1, 9, 3, 3, 1, 0, 1, 3, 3, 9, 1, 9, 9, 1, 9, 3, 3, 1, 0, 1, 3, 3, 9, 1, 9, 9, 1, 9, 3, 3, 1, 0, 1, 3, 3, 9, 1, 9, 9, 1, 9, 3, 3, 1, 0, 1, 3, 3, 9, 1, 9, 9, 1, 9, 3, 3, 1, 0, 1, 3, 3, 9, 1, 9, 9, 1, 9, 3, 3, 1, 0, 1, 3, 3, 9, 1, 9, 9, 1, 9

Offset: 0

N. J. A. Sloane, May 13 2002

Equivalently: n^(12*m + 4) mod 13. - G. C. Greubel, Apr 01 2016

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

[Modexp(n, 4, 13): n in [0..100]]; // Vincenzo Librandi, Apr 02 2016 - after Bruno Berselli in A070403.

PowerMod[Range[0, 100], 4, 13] (* G. C. Greubel, Apr 01 2016 *)

a(n)=n^4%13 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,4,13)for n in range(0, 101)] # Zerinvary Lajos, Oct 31 2009

From G. C. Greubel, Apr 01 2016: (Start)
a(n+13) = a(n).
a(13*m) = 0.
G.f.: (x +3*x^2 +3*x^3 +9*x^4 +x^5 +9*x^6 +9*x^7 +x^8 +9*x^9 +3*x^10 +3*x^11 +x^12)/(1 - x^13). (End)

cp's OEIS Frontend

A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].

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A010689 Periodic sequence: Repeat 1, 8.

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A070366 a(n) = 5^n mod 9.

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A100402 Digital root of 4^n.

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A156760 5*4^n-1.

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A381487 Numbers which are a power of their digital root.

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A144449 a(n) = 4(4 + 9n^2 + 15*n).