A070402 -id:A070402 - cp's OEIS frontend

A000689 Final decimal digit of 2^n.

1, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6

Offset: 0

N. J. A. Sloane

These are the analogs of the powers of 2 in carryless arithmetic mod 10.
Let G = {2,4,8,6}. Let o be defined as XoY = least significant digit in XY. Then (G,o) is an Abelian group wherein 2 is a generator (also see the first comment under A001148). - K.V.Iyer, Mar 12 2010
This is also the decimal expansion of 227/1818. - Kritsada Moomuang, Dec 21 2021

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 6*x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 6*x^8 + ...

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000079, A000855, A008952, A034887, A173635.

a000689 n = a000689_list !! n
a000689_list = 1 : cycle [2,4,8,6]  -- Reinhard Zumkeller, Sep 15 2011

[2^n mod 10: n in [0..150]]; // Vincenzo Librandi, Apr 12 2011

Table[PowerMod[2, n, 10], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)

for(n=0,80, if(n,{x=(n+3)%4+1; print1(10-(4*x^3+47*x-27*x^2)/3,", ")},{print1("1, ")}))

[power_mod(2,n,10)for n in range(0, 81)] # Zerinvary Lajos, Nov 03 2009

Periodic with period 4.
a(n) = 2^n mod 10.
a(n) = A002081(n) - A002081(n-1), for n > 0.
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3), n > 3.
G.f.: (x+3*x^2+5*x^3+1)/((1-x) * (1+x^2)). (End)
For n >= 1, a(n) = 10 - (4x^3 + 47x - 27x^2)/3, where x = (n+3) mod 4 + 1.
For n >= 1, a(n) = A070402(n) + 5*floor( ((n-1) mod 4)/2 ).
G.f.: 1 / (1 - 2*x / (1 + 5*x^3 / (1 + x / (1 - 3*x / (1 + 3*x))))). - Michael Somos, May 12 2012
a(n) = 5 + cos((n*Pi)/2) - 3*sin((n*Pi)/2) for n >= 1. - Kritsada Moomuang, Dec 21 2021

A201908 Irregular triangle of 2^k mod (2n-1).

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Dec 07 2011

The length of the rows is given by A002326. For n > 1, the first term of row n is 1 and the last term is n. Many sequences are in this one: starting at A036117 (mod 11) and A070335 (mod 23).

Row n, for n >= 2, divided elementwise by (2*n-1) gives the cycles of iterations of the doubling function D(x) = 2*x or 2*x-1 if 0 <= x < 1/2 or , 1/2 <= x < 1, respectively, with seed 1/(2*n-1). See the Devaney reference, pp. 25-26. D^[k](x) = frac(2^k/(2*n-1)), for k = 0, 1, ..., A002326(n-1) - 1. E.g., n = 3: 1/5, 2/5, 4/5, 3/5. - Gary W. Adamson and Wolfdieter Lang, Jul 29 2020.

			The irregular triangle T(n, k) begins:
n\k  0 1 2 3  4  5  6  7 8  9 10 11 12 13 14 15 16 17 ...
---------------------------------------------------------
1:   0
2:   1 2
3:   1 2 4 3
4:   1 2 4
5:   1 2 4 8  7  5
6:   1 2 4 8  5 10  9  7 3  6
7:   1 2 4 8  3  6 12 11 9  5 10  7
8:   1 2 4 8
9:   1 2 4 8 16 15 13  9
10:  1 2 4 8 16 13  7 14 9 18 17 15 11  3  6 12  5 10
... reformatted by _Wolfdieter Lang_, Jul 29 2020.

Robert L. Devaney, A First Course in Chaotic Dynamical Systems, Addison-Wesley., 1992. pp. 24-25

T. D. Noe, Rows n = 1..100, flattened

Cf. A002326, A201909 (3^k), A201910 (5^k), A201911 (7^k).

Cf. A000034 (3), A070402 (5), A069705 (7), A036117 (11), A036118 (13), A062116 (17), A036120 (19), A070347 (21), A070335 (23), A070336 (25), A070337 (27), A036122 (29), A070338 (33), A070339 (35), A036124 (37), A070340 (39), A070348 (41), A070349 (43), A070350 (45), A070351 (47), A036128 (53), A036129 (59), A036130 (61), A036131 (67), A036135 (83), A036138 (101), A036140 (107), A201920 (125), A036144 (131), A036146 (139), A036147 (149), A036150 (163), A036152 (173), A036153 (179), A036154 (181), A036157 (197), A036159 (211), A036161 (227).

R:=List([0..72],n->OrderMod(2,2*n+1));;
Flat(Concatenation([0],List([2..11],n->List([0..R[n]-1],k->PowerMod(2,k,2*n-1))))); # Muniru A Asiru, Feb 02 2019

nn = 30; p = 2; t = p^Range[0, nn]; Flatten[Table[If[IntegerQ[Log[p, n]], {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, 1, nn, 2}]]

T(n, k) = 2^k mod (2*n-1), n >= 1, k = 0, 1, ..., A002326(n-1) - 1.

T(n, k) = (2*n-1)*frac(2^k/(2*n-1)), n >= 1, k = 0, 1, ..., A002326(n-1) - 1, with the fractional part frac(x) = x - floor(x). - Wolfdieter Lang, Jul 29 2020

A201912 Irregular triangle of 2^k mod prime(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12, 1, 2, 4, 8, 16, 3, 6, 12, 24

Offset: 1

T. D. Noe, Dec 17 2011

The row lengths are in A014664. For n > 1, the first term of each row is 1 and the last term is 2*prime(n)-1, which is A006254. Many sequences are in this one.

			The first 11 rows are:
2:  0;
3:  1, 2;
5:  1, 2, 4, 3;
7:  1, 2, 4;
11: 1, 2, 4, 8,  5, 10,  9,  7,  3,  6;
13: 1, 2, 4, 8,  3,  6, 12, 11,  9,  5, 10,  7;
17: 1, 2, 4, 8, 16, 15, 13,  9;
19: 1, 2, 4, 8, 16, 13,  7, 14,  9, 18, 17, 15, 11,  3,  6, 12,  5, 10;
23: 1, 2, 4, 8, 16,  9, 18, 13,  3,  6, 12;
29: 1, 2, 4, 8, 16,  3,  6, 12, 24, 19,  9, 18,  7, 14, 28, 27, 25, 21, 13, 26, 23, 17, 5, 10, 20, 11, 22, 15;
31: 1, 2, 4, 8, 16;

T. D. Noe, Rows n = 1..60, flattened

Cf. A062117, A201908, A201909, A201910, A201911.

Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), A269266 (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67), A036135 (p=83), A036138 (p=101), A036140 (p=107), A036144 (p=131), A036146 (p=139), A036147 (p=149), A036150 (p=163), A036152 (p=173), A036153 (p=179), A036154 (p=181), A036157 (p=197), A036159 (p=211), A036161 (p=227).

P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(2,P[n]));;
Flat(Concatenation([0],List([2..10],n->List([0..R[n]-1],k->PowerMod(2,k,P[n]))))); # Muniru A Asiru, Feb 01 2019

nn = 10; p = 2; t = p^Range[0,Prime[nn]]; Flatten[Table[If[Mod[n, p] == 0, {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, Prime[Range[nn]]}]]

A201920 a(n) = 2^n mod 125.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 3, 6, 12, 24, 48, 96, 67, 9, 18, 36, 72, 19, 38, 76, 27, 54, 108, 91, 57, 114, 103, 81, 37, 74, 23, 46, 92, 59, 118, 111, 97, 69, 13, 26, 52, 104, 83, 41, 82, 39, 78, 31, 62, 124, 123, 121, 117, 109, 93, 61, 122, 119, 113, 101, 77, 29

Offset: 0

Franz Vrabec, Dec 06 2011

			a(7) = 2^7 mod 125 = 3.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1, 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, 0, -1, 1).

Cf. A070402, A070336.

a:=List([0..100],n->PowerMod(2,n,125));; Print(a); # Muniru A Asiru, Jan 27 2019

[Modexp(2, n, 125): n in [0..120]]; // G. C. Greubel, Oct 17 2018

PowerMod[2,Range[0,100],125] (* Harvey P. Dale, Aug 12 2013 *)

a(n)=lift(Mod(2,125)^n) \\ Charles R Greathouse IV, Mar 22 2016

For n > 50: a(n) = a(n-1) - a(n-50) + a(n-51).
G.f.: (1 + x + 2x^2 + 4x^3 + 8x^4 + 16x^5 + 32x^6 - 61x^7 + 3x^8 + 6x^9 + 12x^10 + 24x^11 + 48x^12 - 29x^13 - 58x^14 + 9x^15 + 18x^16 + 36x^17 - 53x^18 + 19x^19 + 38x^20 - 49x^21 + 27x^22 + 54x^23 - 17x^24 - 34x^25 + 57x^26 - 11x^27 - 22x^28 - 44x^29 + 37x^30 - 51x^31 + 23x^32 + 46x^33 - 33x^34 + 59x^35 - 7x^36 - 14x^37 - 28x^38 - 56x^39 + 13x^40 + 26x^41 + 52x^42 - 21x^43 - 42x^44 + 41x^45 - 43x^46 + 39x^47 - 47x^48 + 31x^49 + 63x^50) / ((1-x)*(1+x^2)*(1 - x^2 + x^4 - x^6 + x^8 - x^10 + x^12 - x^14 + x^16 - x^18 + x^20 - x^22 + x^24 - x^26 + x^28 - x^30 + x^32 - x^34 + x^36 - x^38 + x^40 - x^42 + x^44 - x^46 + x^48)).
Periodic with period 100.

A269266 a(n) = 2^n mod 31.

Original entry on oeis.org

1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1

Offset: 0

Vincenzo Librandi, Mar 31 2016

Continued fraction expansion of (1651+sqrt(3236405))/2386. - Bruno Berselli, Mar 31 2016

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A201912 (11th row of the triangle).

Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), this sequence (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67).

List([0..70],n->PowerMod(2,n,31)); # Muniru A Asiru, Jan 30 2019

Magma
```
[Modexp(2, n, 31): n in [0..100]];
```

&cat [[1,2,4,8,16]^^20] // Bruno Berselli, Mar 31 2016

Mathematica
```
PowerMod[2, Range[0, 100], 31]
```

a(n)=2^(n%5) \\ Charles R Greathouse IV, Mar 31 2016

x='x+O('x^99); Vec((1+2*x+4*x^2+8*x^3+16*x^4)/(1-x^5)) \\ Altug Alkan, Mar 31 2016

for n in range(0,100):print(2**n%31) # Soumil Mandal, Apr 03 2016

def A269266(n): return pow(2,n,31) # Chai Wah Wu, Jan 03 2022

[2^mod(n,5) for n in (0..100)] # Bruno Berselli, Mar 31 2016

G.f.: (1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4)/(1 - x^5).
a(n) = a(n-5).
a(n) = 2^(n mod 5). - Bruno Berselli, Mar 31 2016

A349767 Numbers m such that 2^m - m is divisible by 5.

Original entry on oeis.org

3, 14, 16, 17, 23, 34, 36, 37, 43, 54, 56, 57, 63, 74, 76, 77, 83, 94, 96, 97, 103, 114, 116, 117, 123, 134, 136, 137, 143, 154, 156, 157, 163, 174, 176, 177, 183, 194, 196, 197, 203, 214, 216, 217, 223, 234, 236, 237, 243, 254, 256, 257, 263, 274, 276, 277, 283, 294, 296, 297, 303

Offset: 1

Bernard Schott, Dec 10 2021

nonn

For every prime p, there are infinitely many numbers m such that 2^m - m (A000325) is divisible by p, here are numbers m corresponding to p = 5.

Equivalently, numbers that are congruent to {3, 14, 16, 17, 23, 34, 36, 37, 43, 54, 56, 57} mod 60, <==> numbers that are congruent to {+-3, +-14, +-16, +-17, +-23, +-34} mod 60.

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1983, page 158, 1993.

The IMO Compendium, Problem 4, 15th Canadian Mathematical Olympiad 1983.
Index to sequences related to Olympiads.

Cf. A000325, A070402, A010874.

Similar with: A299174 (p = 2), A047257 (p = 3), this sequence (p = 5).

filter:= n -> 2^n-n mod 5 = 0 : select(filter, [$1..400]);

Select[Range[300], PowerMod[2, #, 5] == Mod[#, 5] &] (* Amiram Eldar, Dec 10 2021 *)

isok(m) = Mod(2, 5)^m == Mod(m, 5); \\ Michel Marcus, Dec 10 2021

def ok(n): return pow(2, n, 5) == n%5
print([k for k in range(357) if ok(k)]) # Michael S. Branicky, Dec 10 2021

cp's OEIS Frontend

A000689 Final decimal digit of 2^n.

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A201908 Irregular triangle of 2^k mod (2n-1).

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A201912 Irregular triangle of 2^k mod prime(n).

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A201920 a(n) = 2^n mod 125.

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A269266 a(n) = 2^n mod 31.

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A349767 Numbers m such that 2^m - m is divisible by 5.

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