A056969 -id:A056969 - cp's OEIS frontend

A066601 a(n) = 3^n mod n.

0, 1, 0, 1, 3, 3, 3, 1, 0, 9, 3, 9, 3, 9, 12, 1, 3, 9, 3, 1, 6, 9, 3, 9, 18, 9, 0, 25, 3, 9, 3, 1, 27, 9, 12, 9, 3, 9, 27, 1, 3, 15, 3, 37, 18, 9, 3, 33, 31, 49, 27, 29, 3, 27, 12, 9, 27, 9, 3, 21, 3, 9, 27, 1, 48, 3, 3, 13, 27, 39, 3, 9, 3, 9, 57

Offset: 1

Amarnath Murthy, Dec 22 2001

			a(7) = 3 as 3^7 = 2187 = 7*312 + 3.

Harry J. Smith and Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. k^n mod n: A015910 (k=2), this sequence (k=3), A066602 (k=4), A066603 (k=5), A066604 (k=6), A066438 (k=7), A066439 (k=8), A066440 (k=9), A056969 (k=10), A066441 (k=11), A066442 (k=12), A116609 (k=13).

seq(irem(3^n,n),n=1..75); # Zerinvary Lajos, Apr 20 2008

Mathematica
```
Table[PowerMod[3, n, n], {n, 75}]
```

a(n) = { lift(Mod(3, n)^n) } \\ Harry J. Smith, Mar 09 2010

def A066601(n): return pow(3,n,n) # Chai Wah Wu, Aug 24 2023

More terms from Robert G. Wilson v, Dec 27 2001

A066438 a(n) = 7^n mod n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 0, 1, 1, 9, 7, 1, 7, 7, 13, 1, 7, 1, 7, 1, 7, 5, 7, 1, 7, 23, 1, 21, 7, 19, 7, 1, 13, 15, 28, 1, 7, 11, 31, 1, 7, 7, 7, 25, 37, 3, 7, 1, 0, 49, 37, 9, 7, 1, 43, 49, 1, 49, 7, 1, 7, 49, 28, 1, 37, 37, 7, 21, 67, 49, 7, 1, 7, 49, 43, 45, 28, 25, 7, 1

Offset: 1

Robert G. Wilson v, Dec 27 2001

nonn

Harry J. Smith and Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. k^n mod n; A015910 (k=2), A066601 (k=3), A066602 (k=4), A066603 (k=5), A066604 (k=6), this sequence (k=7), A066439 (k=8), A066440 (k=9), A056969 (k=10), A066441 (k=11), A066442 (k=12), A116609 (k=13).

seq(irem(7^n,n),n=1..80); # Zerinvary Lajos, Apr 20 2008

Mathematica
```
Table[PowerMod[7, n, n], {n, 80} ]
```

a(n) = { lift(Mod(7, n)^n) } \\ Harry J. Smith, Feb 14 2010

A066603 a(n) = 5^n mod n.

Original entry on oeis.org

0, 1, 2, 1, 0, 1, 5, 1, 8, 5, 5, 1, 5, 11, 5, 1, 5, 1, 5, 5, 20, 3, 5, 1, 0, 25, 26, 9, 5, 25, 5, 1, 26, 25, 10, 1, 5, 25, 8, 25, 5, 1, 5, 9, 35, 25, 5, 1, 19, 25, 23, 1, 5, 1, 45, 25, 11, 25, 5, 25, 5, 25, 62, 1, 5, 49, 5, 13, 56, 65, 5, 1, 5, 25, 50, 17, 3, 25, 5, 65, 80, 25, 5, 1, 65

Offset: 1

Amarnath Murthy, Dec 22 2001

			a(7) = 5 as 5^7 = 78125 = 7*11160 + 5.

Harry J. Smith and Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. k^n mod n: A015910 (k=2), A066601 (k=3), A066602 (k=4), this sequence (k=5), A066604 (k=6), A066438 (k=7), A066439 (k=8), A066440 (k=9), A056969 (k=10), A066441 (k=11), A066442 (k=12), A116609 (k=13).

seq(irem(5^n,n),n=1..85); # Zerinvary Lajos, Apr 20 2008

Mathematica
```
Table[PowerMod[5, n, n], {n, 85} ]
```

a(n) = { lift(Mod(5, n)^n) } \\ Harry J. Smith, Mar 09 2010

More terms from Robert G. Wilson v, Dec 27 2001

A066441 a(n) = 11^n mod n.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 4, 1, 8, 1, 0, 1, 11, 9, 11, 1, 11, 1, 11, 1, 8, 11, 11, 1, 1, 17, 26, 25, 11, 1, 11, 1, 11, 19, 16, 1, 11, 7, 5, 1, 11, 1, 11, 33, 26, 29, 11, 1, 18, 1, 5, 29, 11, 1, 11, 9, 20, 5, 11, 1, 11, 59, 8, 1, 46, 55, 11, 21, 20, 11, 11, 1, 11, 47, 26, 49, 44, 25, 11, 1

Offset: 1

Robert G. Wilson v, Dec 27 2001

nonn

Harry J. Smith and Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. k^n mod n: A015910 (k=2), A066601 (k=3), A066602 (k=4), A066603 (k=5), A066604 (k=6), A066438 (k=7), A066439 (k=8), A066440 (k=9), A056969 (k=10), this sequence (k=11), A066442 (k=12), A116609 (k=13).

seq(irem(11^n,n),n=1..80); # Zerinvary Lajos, Apr 20 2008

Mathematica
```
Table[PowerMod[11, n, n], {n, 80} ]
```

a(n) = { lift(Mod(11, n)^n) } \\ Harry J. Smith, Feb 14 2010

A060154 Table T(n,k) by antidiagonals of n^k mod k [n,k >= 1].

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Mar 12 2001

			T(5,3) = 5^3 mod 3 = 125 mod 3 = 2.
Rows start:
  0, 1, 1, 1, 1, ...
  0, 0, 2, 0, 2, ...
  0, 1, 0, 1, 3, ...
  0, 0, 1, 0, 4, ...
  0, 1, 2, 1, 0, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Rows include A057427, A015910, A056969.
Columns include A000004, A000035 (several times), A010872, A010874, A010876, A021559 and other periodic sequences.
Diagonals include A000004 and A057427.
Cf. A114448.

T(n, k) = A051129(n, k)-n*A060155(n, k).

A060156 a(n) = floor(10^n/n).

Original entry on oeis.org

10, 50, 333, 2500, 20000, 166666, 1428571, 12500000, 111111111, 1000000000, 9090909090, 83333333333, 769230769230, 7142857142857, 66666666666666, 625000000000000

Offset: 1

Henry Bottomley, Mar 12 2001

nonn

			a(6) = floor(1000000/6) = 166666.

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A000799, A056159, A060155.

{ default(realprecision, 10); for (n=1, 200, write("b060156.txt", n, " ", floor(10^n/n)); ) } \\ Harry J. Smith, Jul 02 2009

a(n) = (A011557(n) - A056969(n))/n.

A121912 Numbers k such that 10^k == 10 (mod k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 15, 17, 18, 19, 23, 29, 30, 31, 33, 37, 41, 43, 45, 47, 53, 55, 59, 61, 67, 71, 73, 79, 83, 89, 90, 91, 97, 99, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 165, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Zak Seidov, Sep 02 2006

nonn

By Fermat, all primes are members.
Numbers k not divisible by 4 or 25 such that the multiplicative order of 10 mod (k/gcd(k,10)) divides k-1. - Robert Israel, Feb 10 2019
10^2^k + 1, 10^5^k + 1 and 10^10^k + 1 are terms for k >= 0. - Jinyuan Wang, Feb 11 2019

			13 is a term because 10^13 = 13*769230769230 + 10.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A056969 (10^n modulo n), A121014 (Nonprime terms in A121912).

filter:= n -> (10 &^ n - 10 mod n = 0):
select(filter, [$1..1000]); # Robert Israel, Feb 10 2019

Select[Range[250], PowerMod[10, #, # ] == Mod[10, # ] &] (* Ray Chandler, Sep 02 2006 *)

is(n) = Mod(10, n)^n == Mod(10, n) \\ Jinyuan Wang, Feb 11 2019

A056968 10^(n-1) modulo n.

Original entry on oeis.org

0, 0, 1, 0, 0, 4, 1, 0, 1, 0, 1, 4, 1, 10, 10, 0, 1, 10, 1, 0, 16, 10, 1, 16, 0, 10, 19, 20, 1, 10, 1, 0, 1, 10, 25, 28, 1, 10, 22, 0, 1, 40, 1, 32, 10, 10, 1, 16, 8, 0, 49, 12, 1, 46, 45, 24, 43, 10, 1, 40, 1, 10, 37, 0, 55, 10, 1, 48, 31, 20, 1, 64, 1, 10, 25, 12, 67, 4, 1, 0, 73, 10, 1

Offset: 1

Henry Bottomley, Jul 20 2000

			a(6)=4 since 100000=6*16666+4

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003592, A053041, A056969.

0, seq(10&^(n-1) mod n, n=2..100); # Robert Israel, Nov 25 2024

Table[PowerMod[10,n-1,n],{n,100}] (* Harvey P. Dale, Jul 17 2021 *)

If n is of form 2^i*5^j then a(n)=0, otherwise a(n)=10^(n-1)+n-A053041(n)
From Robert Israel, Nov 25 2024: (Start)
If n is prime other than 2 or 5, then a(n) = 1.
If n = 2^i * 5^j * p where p is a prime > 10^(2^i * 5^j), then a(n) = 10^(2^i * 5^j).
If n = 2^i * 5^j * p where p is a prime and
  2^(2^i * 5^j - 1 - i) * 5^(2^i * 5^j -1 - j) > p > 2^(2^i * 5^j-2 - u) * 5^(2^i * 5^j-1-j),
then a(n) =  10^(2^i * 5^j - 1) - 2^i * 5^j * p.
For example, with i = 0 and j = 1 we get a(5*p) = 10^4 - 5*p if p is a prime between 1000 and 2000.
(End)

A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Feb 14 2006

Alternate description: triangular array a(n, k) = n^k (mod k) read by rows (n > 1, 0 < k < n). This is equivalent because a(n, k) = a(n-k, k). - David Wasserman, Jan 25 2007

			2^6 = 64 and 64 (mod 6) is 4. So a(2,6) = 4.

Cf. A051127, A051128, A094263. Transpose of A060154. First 13 rows are A057427(n-1), A015910, A066601, A066602, A066603, A066604, A066438, A066439, A066440, A056969, A066441, A066442, A116609. First 12 columns are A000004, A000035, A010872, A000035, A010874, A070431, A010876, A000035, A021559(n+1), A008959, A010880, A070435.

a[n_, k_] := Mod[n^k, k]; Table[a[n - k + 1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

More terms from David Wasserman, Jan 25 2007

A293907 Numbers n for which 10^n mod n = 2^k for some positive integer k.

Original entry on oeis.org

6, 12, 14, 24, 28, 34, 46, 48, 52, 56, 68, 72, 84, 92, 96, 112, 117, 123, 126, 136, 144, 168, 186, 192, 204, 208, 224, 228, 249, 252, 266, 272, 288, 328, 336, 356, 372, 384, 392, 408, 416, 448, 464, 488, 498, 504, 516

Offset: 1

Björn Ch. Buchli, Oct 19 2017

nonn

Odd numbers in this sequence: 117, 123, 249, 747, 4043, 5031, 11573, 12129, 14481, 29489, 34719, 35549, 84123, 124631, 173329, 217391, 266799, 458523, 472173, 490561, 551759, 658499, 675431, 721773, 800397, 1375569, 1917843, 2300079, 3194787, 3394893, 4236747, 5031039, 5043957, 5169333, ....

			For n = 6, 10^6 mod 6 = 4 = 2^2;
For n = 14, 10^14 mod 14 = 2 = 2^1;
For n = 84, 10^84 mod 84 = 64 = 2^6;
For n = 272, 10^272 mod 272 = 256 = 2^8.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A056969 (10^n modulo n).

pm2Q[n_]:=Module[{c=PowerMod[10,n,n]},c>1&&IntegerQ[Log2[c]]]; Select[ Range[600],pm2Q] (* Harvey P. Dale, Mar 29 2023 *)

is(n)=my(k=lift(Mod(10,n)^n)); k>1 && k>>valuation(k,2)==1 \\ Charles R Greathouse IV, Oct 19 2017

More terms from Michel Marcus, Oct 19 2017

cp's OEIS Frontend

A066601 a(n) = 3^n mod n.

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A066438 a(n) = 7^n mod n.

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A066603 a(n) = 5^n mod n.

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A066441 a(n) = 11^n mod n.

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A060154 Table T(n,k) by antidiagonals of n^k mod k [n,k >= 1].

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A060156 a(n) = floor(10^n/n).

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A121912 Numbers k such that 10^k == 10 (mod k).

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Maple

Mathematica

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A056968 10^(n-1) modulo n.

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