A066603 -id:A066603 - 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

A056969 a(n) = 10^n modulo n.

Original entry on oeis.org

0, 0, 1, 0, 0, 4, 3, 0, 1, 0, 10, 4, 10, 2, 10, 0, 10, 10, 10, 0, 13, 12, 10, 16, 0, 22, 1, 4, 10, 10, 10, 0, 10, 32, 5, 28, 10, 24, 25, 0, 10, 22, 10, 12, 10, 8, 10, 16, 31, 0, 31, 16, 10, 28, 10, 16, 31, 42, 10, 40, 10, 38, 55, 0, 30, 34, 10, 4, 34, 60, 10, 64, 10, 26, 25, 44, 54, 40

Offset: 1

Henry Bottomley, Jul 20 2000

nonn

			a(7) = 3 since 10000000 = 7*1428571+3

Seiichi Manyama, Table of n, a(n) for n = 1..10000

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), this sequence (k=10), A066441 (k=11), A066442 (k=12), A116609 (k=13).

seq(irem(10^n,n),n=1..78); # Zerinvary Lajos, Apr 20 2008

Mathematica
```
Table[PowerMod[10, n, n], {n, 80} ]
```

a(n) = lift(Mod(10, n)^n); \\ Michel Marcus, Oct 19 2017

a(n) = 10*A056968(n) mod n = A011557(n) mod n.

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

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

A277348 Positive integers n such that n | (5^n + 6).

Original entry on oeis.org

1, 11, 341, 581337017, 7202608727, 27146455379, 1358496201131, 9843739213499, 172392038905691

Offset: 1

Seiichi Manyama, Oct 10 2016

No other terms below 10^15. - Max Alekseyev, Oct 17 2016

			5^11 + 6 = 48828131 = 11 * 4438921, so 11 is a term.

Cf. A066603.

Cf. Solutions to 5^n == k (mod n): this sequence (k=-6), A015891 (k=-5), A123047 (k=-4), A123052 (k=-3), A123062 (k=-2), A015951 (k=-1), A067946 (k=1), A124246 (k=2), A123061 (k=3), A125949 (k=4), A123091 (k=5), A277350 (k=6).

isok(n) = Mod(5, n)^n == -6; \\ Michel Marcus, Oct 10 2016

A066603(a(n)) = a(n) - 6 for n > 1.

a(5)-a(9) from Max Alekseyev, Oct 17 2016

A096385 a(n) = smallest prime p with p^n mod n = 1.

Original entry on oeis.org

3, 7, 3, 11, 5, 29, 3, 7, 11, 23, 5, 53, 13, 31, 3, 103, 5, 191, 3, 37, 23, 47, 5, 11, 53, 7, 13, 59, 11, 311, 3, 67, 67, 71, 5, 149, 37, 61, 3, 83, 5, 173, 23, 31, 47, 283, 5, 29, 11, 103, 5, 107, 5, 31, 13, 7, 59, 709, 7, 367, 61, 37, 3, 131, 23, 269, 13, 139, 29, 569

Offset: 2

Reinhard Zumkeller, Aug 05 2004

nonn

			n=5: 2^5=32=5*6+2, 3^5=243=5*48+3, 5^5 mod 5 = 0, 7^5=16807=5*3361+2, 11^5=161051=5*32210+1: a(5)=11.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A015910, A066601, A066603, A066438, A066441.

With[{prs=Prime[Range[200]]},Table[SelectFirst[prs,PowerMod[#,n,n]==1&],{n,2,80}]] (* The program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, Aug 31 2015 *)

a(n) = my(p=2); while (Mod(p,n)^n !=1, p=nextprime(p+1)); p; \\ Michel Marcus, Feb 07 2021

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

A297967 a(n) = 5^n mod prime(n).

Original entry on oeis.org

1, 1, 0, 2, 1, 12, 10, 4, 11, 20, 25, 10, 39, 36, 41, 13, 36, 58, 52, 1, 17, 26, 19, 64, 13, 5, 94, 14, 25, 36, 40, 74, 117, 81, 6, 123, 24, 155, 134, 117, 20, 42, 69, 36, 185, 111, 121, 16, 206, 159, 42, 220, 47, 123, 130, 61, 57, 83, 6, 79, 270, 14, 91

Offset: 1

Vincenzo Librandi, Jan 15 2018

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040, A000351, A064367, A066603, A297966.

[Modexp(5, n, NthPrime(n)): n in [1..80]];

Mathematica
```
Array[PowerMod[5, #, Prime@#]&, 80]
```

a(n) = A000351(n) mod A000040(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Extensions

A056969 a(n) = 10^n modulo n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A277348 Positive integers n such that n | (5^n + 6).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A096385 a(n) = smallest prime p with p^n mod n = 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A297967 a(n) = 5^n mod prime(n).

Views

Author