A066602 -id:A066602 - 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

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

A251603 Numbers k such that k + 2 divides k^k - 2.

Original entry on oeis.org

3, 4551, 46775, 82503, 106976, 1642796, 4290771, 4492203, 4976427, 21537831, 21549347, 21879936, 51127259, 56786087, 60296571, 80837771, 87761787, 94424463, 96593696, 138644871, 168864999, 221395539, 255881451, 297460451, 305198247, 360306363, 562654203

Offset: 1

Juri-Stepan Gerasimov, Dec 05 2014

nonn

Numbers k such that (k^k - 2)/(k + 2) is an integer.
Since k == -2 (mod k+2), also numbers k such that k + 2 divides (-2)^k - 2. - Robert Israel, Jan 04 2015
Numbers k == 0 (mod 4) such that A066602(k/2+1) = 8, and odd numbers k such that k = 3 or A082493(k+2) = 8. - Robert Israel, Apr 08 2015

			3 is in this sequence because 3 + 2 = 5 divides 3^3 - 2 = 25.

Max Alekseyev, Table of n, a(n) for n = 1..890 (all terms below 10^15)

Cf. A001477, A004273, A004275, A066602, A082493, A081765, A213382, A252606, A357125.

[n: n in [0..10000] | Denominator((n^n-2)/(n+2)) eq 1];

isA251603 := proc(n)
    if modp(n &^ n-2,n+2) = 0 then
        true;
    else
        false;
    end if;
end proc:
A251603 := proc(n)
    option remember;
    local a;
    if n = 1 then
        3;
    else
        for a from procname(n-1)+1 do
            if isA251603(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jan 09 2015

Select[Range[10^6], Mod[PowerMod[#, #, # + 2] - 2, # + 2] == 0 &] (* Michael De Vlieger, Dec 20 2014, based on Robert G. Wilson v at A252041 *)

for(n=1,10^9,if(Mod(n,n+2)^n==+2,print1(n,", "))); \\ Joerg Arndt, Dec 06 2014

A251603_list = [n for n in range(1,10**6) if pow(n, n, n+2) == 2] # Chai Wah Wu, Apr 13 2015

The even terms form A122711, the odd terms are those in A245319 (forming A357125) decreased by 2. - Max Alekseyev, Sep 22 2016

a(6)-a(27) from Joerg Arndt, Dec 06 2014

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

cp's OEIS Frontend

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

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Maple

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

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Maple

Mathematica

PARI

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

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Maple

Mathematica

PARI

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

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Maple

Mathematica

PARI

Extensions

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

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Programs

Maple

Mathematica

PARI

A251603 Numbers k such that k + 2 divides k^k - 2.

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Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

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

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Author

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