A008960 -id:A008960 - cp's OEIS frontend

A010879 Final digit of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0

Offset: 0

N. J. A. Sloane

Also decimal expansion of 137174210/1111111111 = 0.1234567890123456789012345678901234... - Jason Earls, Mar 19 2001
In general the base k expansion of A062808(k)/A048861(k) (k>=2) will produce the numbers 0,1,2,...,k-1 repeated with period k, equivalent to the sequence n mod k. The k-digit number in base k 123...(k-1)0 (base k) expressed in decimal is A062808(k), whereas A048861(k) = k^k-1. In particular, A062808(10)/A048861(10)=1234567890/9999999999=137174210/1111111111.
a(n) = n^5 mod 10. - Zerinvary Lajos, Nov 04 2009

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).
Index entries for sequences related to final digits of numbers

Cf. A034948, A059988, A048861, A062808, A086457, A086458.
Cf. A008959, A008960, A070514. - Doug Bell, Jun 15 2015
Partial sums: A130488. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486, A130487.

a010879 = (`mod` 10)
a010879_list = cycle [0..9]  -- Reinhard Zumkeller, Mar 26 2012

[n mod(10): n in [0..90]]; // Vincenzo Librandi, Jun 17 2015

A010879 := proc(n)
    n mod 10 ;
end proc: # R. J. Mathar, Jul 12 2013

Table[10*FractionalPart[n/10], {n, 1, 300}] (* Enrique Pérez Herrero, Jul 30 2009 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{0, 1, 2, 3, 4, 5, 6, 7, 8, 9},81] (* Ray Chandler, Aug 26 2015 *)
PadRight[{},100,Range[0,9]] (* Harvey P. Dale, Oct 04 2021 *)

a(n)=n%10 \\ Charles R Greathouse IV, Jun 16 2011

def a(n): return n % 10 # Martin Gergov, Oct 17 2022

[power_mod(n,5,10)for n in range(0, 81)] # Zerinvary Lajos, Nov 04 2009

a(n) = n mod 10.
Periodic with period 10.
From Hieronymus Fischer, May 31 and Jun 11 2007: (Start)
Complex representation: a(n) = 1/10*(1-r^n)*sum{1<=k<10, k*product{1<=m<10,m<>k, (1-r^(n-m))}} where r=exp(Pi/5*i) and i=sqrt(-1).
Trigonometric representation: a(n) = (256/5)^2*(sin(n*Pi/10))^2 * sum{1<=k<10, k*product{1<=m<10,m<>k, (sin((n-m)*Pi/10))^2}}.
G.f.: g(x) = (Sum_{k=1..9} k*x^k)/(1-x^10) = -x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8) / ( (x-1) *(1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) ).
Also: g(x) = x*(9*x^10-10*x^9+1)/((1-x^10)*(1-x)^2).
a(n) = n mod 2+2*(floor(n/2)mod 5) = A000035(n) + 2*A010874(A004526(n)).
Also: a(n) = n mod 5+5*(floor(n/5)mod 2) = A010874(n)+5*A000035(A002266(n)). (End)
a(n) = 10*{n/10}, where {x} means fractional part of x. - Enrique Pérez Herrero, Jul 30 2009
a(n) = n - 10*A059995(n). - Reinhard Zumkeller, Jul 26 2011
a(n) = n^k mod 10, for k > 0, where k mod 4 = 1. - Doug Bell, Jun 15 2015

Formula section edited for better readability by Hieronymus Fischer, Jun 13 2012

A008959 Final digit of squares: a(n) = n^2 mod 10.

Original entry on oeis.org

0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0

Offset: 0

N. J. A. Sloane, Mar 15 1996

a(m*n) = a(m)*a(n) mod 10; a(5*n+k) = a(5*n-k) for k <= 5*n. - Reinhard Zumkeller, Apr 24 2009
a(n) = n^6 mod 10. - Zerinvary Lajos, Nov 06 2009
a(n) = A002015(n) mod 10 = A174452(n) mod 10. - Reinhard Zumkeller, Mar 21 2010
Decimal expansion of 166285490/1111111111. - Alexander R. Povolotsky, Mar 09 2013

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A000290, A070431, A070435, A070438, A070442, A070452, A159852, A010879, A008960, A070514.

[0] cat [Intseq(n^2)[1]: n in [1..80]]; // Bruno Berselli, Feb 14 2013

[n^2 - 10*Floor(n^2/10): n in [0..80]]; // Vincenzo Librandi, Jun 16 2015

A008959:=n->(n^2 mod 10); seq(A008959(n), n=0..50); # Wesley Ivan Hurt, Jun 06 2014

Table[Mod[n^2,10],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2011 *)
PowerMod[Range[0,80],2,10] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,0,1},{0,1,4,9,6,5,6,9,4,1},120] (* Harvey P. Dale, Oct 16 2012 *)

a(n)=n^2%10 \\ Charles R Greathouse IV, Sep 24 2015

[power_mod(n,2,10) for n in range(0, 81)] # Zerinvary Lajos, Nov 06 2009

Periodic with period 10. - Franklin T. Adams-Watters, Mar 13 2006
a(n) = 4.5 - (1 + 5^(1/2))*cos(Pi*n/5) + (-1 - 3/5*5^(1/2))*cos(2*Pi*n/5) + (5^(1/2) - 1)*cos(3*Pi*n/5) + (-1 + 3/5*5^(1/2))*cos(4*Pi*n/5) - 0.5*(-1)^n. - Richard Choulet, Dec 12 2008
a(n) = A010879(A000290(n)). - Reinhard Zumkeller, Jan 04 2009
G.f.: (x^9+4*x^8+9*x^7+6*x^6+5*x^5+6*x^4+9*x^3+4*x^2+x)/(-x^10+1). - Colin Barker, Aug 14 2012
a(n) = n^2 - 10*floor(n^2/10). - Wesley Ivan Hurt, Jun 12 2013
a(n) = (n - 5*A002266(n + 2))^2 + 5*(5*A002266(n + 2) mod 2). - Wesley Ivan Hurt, Jun 06 2014
a(n) = A033569(n+3) mod 10. - Wesley Ivan Hurt, Dec 06 2014
a(n) = n^k mod 10; for k > 0 where k mod 4 = 2. - Doug Bell, Jun 15 2015

A070514 Final digit of n^4: a(n) = n^4 mod 10.

Original entry on oeis.org

0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 0

Offset: 0

N. J. A. Sloane, May 13 2002

Decimal expansion of 538853870/3333333333. - Alexander R. Povolotsky, Mar 09 2013

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

Cf. A010879, A008959, A008960. - Doug Bell, Jun 15 2015

[n^4 mod (10): n in [0..80]]; // Vincenzo Librandi, Jun 16 2015

A070514:=n->n^4 mod 10: seq(A070514(n), n=0..100); # Wesley Ivan Hurt, Apr 01 2016

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 1, 6, 1, 6, 5, 6, 1, 6, 1}, 100] (* Vincenzo Librandi, Jun 16 2015 *)
PowerMod[Range[0, 100], 4, 10] (* G. C. Greubel, Apr 01 2016 *)

vector(100,n,n--;n^4%10) \\ Derek Orr, Jun 16 2015

[power_mod(n,4,10)for n in range(0, 101)] # Zerinvary Lajos, Oct 30 2009

a(n) = n^k mod 10; for k > 0 where k mod 4 = 0. - Doug Bell, Jun 15 2015
From G. C. Greubel, Apr 01 2016: (Start)
a(n) = a(n-10).
a(2*n) = 6*A011558(n).
G.f.: (x +6*x^2 +x^3 +6*x^4 +5*x^5 +6*x^6 +x^7 +6*x^8 +x^9)/(1 - x^10). (End)

A070473 a(n) = n^3 mod 11.

Original entry on oeis.org

0, 1, 8, 5, 9, 4, 7, 2, 6, 3, 10, 0, 1, 8, 5, 9, 4, 7, 2, 6, 3, 10, 0, 1, 8, 5, 9, 4, 7, 2, 6, 3, 10, 0, 1, 8, 5, 9, 4, 7, 2, 6, 3, 10, 0, 1, 8, 5, 9, 4, 7, 2, 6, 3, 10, 0, 1, 8, 5, 9, 4, 7, 2, 6, 3, 10, 0, 1, 8, 5, 9, 4, 7, 2, 6, 3, 10, 0, 1, 8, 5, 9, 4, 7, 2, 6, 3, 10, 0, 1, 8, 5, 9, 4, 7, 2, 6, 3, 10

Offset: 0

N. J. A. Sloane, May 12 2002

Vincenzo Librandi, 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, 1).

Cf. A008960, A070474, A167176.

[n^3 mod 11: n in [0..80] ]; // Vincenzo Librandi, Jun 01 2011

Table[Mod[n^3,11],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2011 *)
PowerMod[Range[0,120],3,11] (* or *) PadRight[{},120,{0,1,8,5,9,4,7,2,6,3,10}] (* Harvey P. Dale, May 04 2019 *)

a(n)=n^3%11 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,3,11 ) for n in range(0, 99)] # Zerinvary Lajos, Oct 29 2009

a(n) = a(n-11). - G. C. Greubel, Mar 26 2016

A070479 a(n) = n^3 mod 17.

Original entry on oeis.org

0, 1, 8, 10, 13, 6, 12, 3, 2, 15, 14, 5, 11, 4, 7, 9, 16, 0, 1, 8, 10, 13, 6, 12, 3, 2, 15, 14, 5, 11, 4, 7, 9, 16, 0, 1, 8, 10, 13, 6, 12, 3, 2, 15, 14, 5, 11, 4, 7, 9, 16, 0, 1, 8, 10, 13, 6, 12, 3, 2, 15, 14, 5, 11, 4, 7, 9, 16, 0, 1, 8, 10, 13, 6, 12, 3, 2, 15, 14, 5, 11, 4, 7, 9, 16, 0, 1

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, 1).

Cf. A008960.

[n^3 mod 17: n in [0..80]]; // Vincenzo Librandi, Jun 19 2014

Table[Mod[n^3, 17], {n, 0, 100}] (* Vincenzo Librandi, Jun 19 2014 *)
PowerMod[Range[0,120],3,17] (* or *) PadRight[{},120,{0,1,8,10,13,6,12,3,2,15,14,5,11,4,7,9,16}] (* Harvey P. Dale, Nov 20 2024 *)

a(n)=n^3%17 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,3,17 )for n in range(0, 87)] # Zerinvary Lajos, Oct 29 2009

From G. C. Greubel, Mar 28 2016: (Start)
a(n) = a(n-17).
G.f.: (-x - 8x^2 - 10x^3 - 13x^4 - 6x^5 - 12x^6 - 3x^7 - 2x^8 - 15x^9 - 14x^10 - 5x^11 - 11x^12 - 4x^13 - 7x^14 - 9x^15 - 16x^16)/(-1 + x^17). (End)

A070481 a(n) = n^3 mod 19.

Original entry on oeis.org

0, 1, 8, 8, 7, 11, 7, 1, 18, 7, 12, 1, 18, 12, 8, 12, 11, 11, 18, 0, 1, 8, 8, 7, 11, 7, 1, 18, 7, 12, 1, 18, 12, 8, 12, 11, 11, 18, 0, 1, 8, 8, 7, 11, 7, 1, 18, 7, 12, 1, 18, 12, 8, 12, 11, 11, 18, 0, 1, 8, 8, 7, 11, 7, 1, 18, 7, 12, 1, 18, 12, 8, 12, 11, 11, 18, 0, 1, 8, 8, 7, 11, 7, 1, 18

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, 1).

Cf. A008960.

[Modexp(n,3,19): n in [0..100]]; // Vincenzo Librandi, Mar 28 2016

Table[Mod[n^3, 19], {n, 0, 100}] (* Vincenzo Librandi, Jun 19 2014 *)
PowerMod[Range[0,90],3,19] (* Harvey P. Dale, Feb 25 2015 *)

a(n)=n^3%19 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,3,19 )for n in range(0, 85)] # Zerinvary Lajos, Oct 29 2009

From G. C. Greubel, Mar 28 2016: (Start)
a(n) = a(n-19).
G.f.: (-x -8*x^2 -8*x^3 -7*x^4 -11*x^5 -7*x^6 -x^7 -18*x^8 -7*x^9 - 12*x^10 -x^11 -18*x^12 -12*x^13 -8*x^14 -12*x^15 -11*x^16 -11*x^17 - 18*x^18)/(-1 + x^19). (End)

A070485 a(n) = n^3 mod 23.

Original entry on oeis.org

0, 1, 8, 4, 18, 10, 9, 21, 6, 16, 11, 20, 3, 12, 7, 17, 2, 14, 13, 5, 19, 15, 22, 0, 1, 8, 4, 18, 10, 9, 21, 6, 16, 11, 20, 3, 12, 7, 17, 2, 14, 13, 5, 19, 15, 22, 0, 1, 8, 4, 18, 10, 9, 21, 6, 16, 11, 20, 3, 12, 7, 17, 2, 14, 13, 5, 19, 15, 22, 0, 1, 8, 4, 18, 10, 9, 21, 6, 16, 11, 20, 3

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, 1).

Cf. A008960.

[n^3 mod 23: n in [0..80]]; // Vincenzo Librandi, Jun 19 2014

Table[Mod[n^3, 23], {n, 0, 100}] (* Vincenzo Librandi, Jun 19 2014 *)
PowerMod[Range[0, 90], 3, 23] (* G. C. Greubel, Mar 28 2016 *)

a(n)=n^3%23 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,3,23 )for n in range(0, 82)] #Zerinvary Lajos, Oct 29 2009

From G. C. Greubel, Mar 28 2016: (Start)
a(n) = a(n-23).
G.f.: (-x -8*x^2 -4*x^3 -18*x^4 -10*x^5 -9*x^6 -21*x^7 -6*x^8 -16*x^9 -11* x^10 -20*x^11 -3*x^12 -12*x^13 -7*x^14 -17*x^15 -2*x^16 -14*x^17 -13*x^18 -5*x^19 -19*x^20 -15*x^21 -22*x^22)/(-1 + x^23). (End)

A070491 a(n) = n^3 mod 29.

Original entry on oeis.org

0, 1, 8, 27, 6, 9, 13, 24, 19, 4, 14, 26, 17, 22, 18, 11, 7, 12, 3, 15, 25, 10, 5, 16, 20, 23, 2, 21, 28, 0, 1, 8, 27, 6, 9, 13, 24, 19, 4, 14, 26, 17, 22, 18, 11, 7, 12, 3, 15, 25, 10, 5, 16, 20, 23, 2, 21, 28, 0, 1, 8, 27, 6, 9, 13, 24, 19, 4, 14, 26, 17, 22, 18, 11, 7, 12, 3, 15

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, 1).

Cf. A008960.

Table[Mod[n^3, 29], {n, 0, 100}] (* Vincenzo Librandi, Jun 19 2014 *)

a(n)=n^3%29 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,3,29 ) for n in range(0, 78)] # Zerinvary Lajos, Oct 30 2009

a(n) = a(n-29). - G. C. Greubel, Mar 30 2016

A070493 a(n) = n^3 mod 31.

Original entry on oeis.org

0, 1, 8, 27, 2, 1, 30, 2, 16, 16, 8, 29, 23, 27, 16, 27, 4, 15, 4, 8, 2, 23, 15, 15, 29, 1, 30, 29, 4, 23, 30, 0, 1, 8, 27, 2, 1, 30, 2, 16, 16, 8, 29, 23, 27, 16, 27, 4, 15, 4, 8, 2, 23, 15, 15, 29, 1, 30, 29, 4, 23, 30, 0, 1, 8, 27, 2, 1, 30, 2, 16, 16, 8, 29, 23, 27, 16, 27, 4, 15, 4

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, 1).

Cf. A008960.

[n^3 mod 31: n in [0..80]]; // Vincenzo Librandi, Jun 19 2014

Table[Mod[n^3, 31], {n, 0, 100}] (* Vincenzo Librandi, Jun 19 2014 *)
PowerMod[Range[0,100],3,31] (* Harvey P. Dale, Apr 13 2015 *)

a(n)=n^3%31 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,3,31 )for n in range(0, 81)] # Zerinvary Lajos, Oct 30 2009

a(n) = a(n-31). - G. C. Greubel, Mar 30 2016

A070499 a(n) = n^3 mod 37.

Original entry on oeis.org

0, 1, 8, 27, 27, 14, 31, 10, 31, 26, 1, 36, 26, 14, 6, 8, 26, 29, 23, 14, 8, 11, 29, 31, 23, 11, 1, 36, 11, 6, 27, 6, 23, 10, 10, 29, 36, 0, 1, 8, 27, 27, 14, 31, 10, 31, 26, 1, 36, 26, 14, 6, 8, 26, 29, 23, 14, 8, 11, 29, 31, 23, 11, 1, 36, 11, 6, 27, 6, 23, 10, 10, 29, 36, 0, 1

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, 1).

Cf. A008960.

[n^3 mod 37: n in [0..80]]; // Vincenzo Librandi, Jun 19 2014

PowerMod[Range[0,80],3,37] (* Harvey P. Dale, May 21 2014 *)
Table[Mod[n^3, 37], {n, 0, 100}] (* Vincenzo Librandi, Jun 19 2014 *)

a(n)=n^3%37 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,3,37 )for n in range(0, 76)] # Zerinvary Lajos, Oct 30 2009

a(n) = a(n-37). - G. C. Greubel, Mar 30 2016

cp's OEIS Frontend

A010879 Final digit of n.

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A008959 Final digit of squares: a(n) = n^2 mod 10.

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A070514 Final digit of n^4: a(n) = n^4 mod 10.

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A070473 a(n) = n^3 mod 11.

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A070479 a(n) = n^3 mod 17.

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A070481 a(n) = n^3 mod 19.

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