A086458 -id:A086458 - 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

A086457 Both n and n^2 have the same initial digit and also n and n^2 have the same final digit when expressed in base 10.

Original entry on oeis.org

0, 1, 10, 11, 95, 96, 100, 101, 105, 106, 110, 111, 115, 116, 120, 121, 125, 126, 130, 131, 135, 136, 140, 141, 895, 896, 950, 951, 955, 956, 960, 961, 965, 966, 970, 971, 975, 976, 980, 981, 985, 986, 990, 991, 995, 996, 1000, 1001, 1005, 1006, 1010, 1011

Offset: 1

Jeremy Gardiner, Jul 20 2003

All terms of A045953 appear in this sequence.
Subsequence of A008851; A045953 and A046851 are subsequences. [Reinhard Zumkeller, Jul 27 2011]
Intersection of A008851 and A089951. - Michel Marcus, Mar 19 2015

			a(12) = 115 appears in the sequence because 115*115 = 13225.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A045953, A086458.

left$(str$(n), 1) = left$(str$(n^2), 1) AND right$(str$(n), 1) = right$(str$(n^2), 1)

a086457 n = a086457_list !! (n-1)
a086457_list = filter (\x -> a000030 x == a000030 (x^2) &&
                             a010879 x == a010879 (x^2)) [0..]
-- Reinhard Zumkeller, Jul 27 2011

ldQ[n_]:=Module[{idn=IntegerDigits[n],idn2=IntegerDigits[n^2]}, First[ idn] == First[idn2]&&Last[idn]==Last[idn2]]; Select[Range[ 0,1100], ldQ]  (* Harvey P. Dale, Feb 06 2011 *)

A000030(a(n)) = A000030(a(n)^2) and A010879(a(n)) = A010879(a(n)^2).

Offset corrected by Reinhard Zumkeller, Jul 27 2011

A179856 Numbers n such that n^3 can be obtained from n by inserting internal (but not necessarily contiguous) digits.

Original entry on oeis.org

10, 11, 29, 34, 99, 100, 101, 106, 109, 110, 114, 119, 120, 124, 125, 274, 275, 276, 279, 281, 290, 296, 299, 314, 315, 316, 319, 320, 324, 325, 329, 330, 335, 336, 340, 966, 970, 975, 976, 979, 986, 990, 996, 999, 1000, 1001, 1004, 1005, 1006, 1010, 1020, 1021, 1024, 1025, 1034, 1049, 1051

Offset: 1

Jonathan Vos Post, Jan 28 2011

This is to A046851 as cubes A000578 are to squares A000290. A subset of A086458 (but note that, i.e., 104^3 = 1124864 starts and ends with the same digits as 104, but lacks an internal "0"). If we require the inserted digits to fill contiguous places, another sequence results, which does not contain 106, for example.

			34^3 = 39304 (insert "930" into "34").
106^3 = 1191016 (insert "191" and "1" into "106").

Cf. A000578, A046851, A086458.

A000030 := proc(n) if n= 0 then 0; else op(-1,convert(n,base,10)) ; end if; end proc:
A010879 := proc(n) n mod 10 ; end proc:
isA086458 := proc(n) A000030(n) = A000030(n^3) and A010879(n) = A010879(n^3) ; end proc:
subsI := proc(c,L) for i from 1 to nops(L) do if op(i,L) = c then return i; end if; end do; return -1 ; end proc:
isSubS := proc(Sub,Sup) if nops(Sub) = 1 then if subsI(op(1,Sub),Sup) > 0 then return true; else return false; end if; elif nops(Sub) = 0 then return true; else f := subsI(op(1,Sub),Sup) ; if f < 0 then return false; else procname( subsop(1=NULL,Sub), [op(f+1..nops(Sup),Sup)] ) ; end if; end if; end proc:
isA179856 := proc(n) if isA086458(n) then dgsn := convert(n,base,10) ; dgsn := op(2..nops(dgsn)-1,dgsn) ; dgsn3 := convert(n^3,base,10) ; dgsn3 := op(2..nops(dgsn3)-1,dgsn3) ; isSubS([dgsn],[dgsn3]) ; else false; end if; end proc:
for n from 10 to 1400 do if isA179856(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Jan 30 2011

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A010879 Final digit of n.

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A086457 Both n and n^2 have the same initial digit and also n and n^2 have the same final digit when expressed in base 10.

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Haskell

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A179856 Numbers n such that n^3 can be obtained from n by inserting internal (but not necessarily contiguous) digits.

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Author

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Examples

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Programs

Maple