A066606 -id:A066606 - cp's OEIS frontend

A213859 a(n) = 2^n mod (n+2).

1, 2, 0, 3, 4, 4, 0, 2, 6, 6, 4, 7, 8, 2, 0, 9, 16, 10, 4, 2, 12, 12, 16, 8, 14, 20, 4, 15, 16, 16, 0, 2, 18, 22, 16, 19, 20, 2, 24, 21, 16, 22, 4, 38, 24, 24, 16, 32, 6, 2, 4, 27, 34, 52, 8, 2, 30, 30, 4, 31, 32, 2, 0, 8, 16, 34, 4, 2, 46, 36, 16, 37, 38, 17

Offset: 0

Alex Ratushnyak, Jun 22 2012

Conjectures:
Indices of zeros: 2^(x+2)-2, x >= 0.
There are infinitely many n's such that a(n)=n.
Every integer k >= 0 appears in a(n) at least once.
Every k >= 0 appears in a(n) infinitely many times.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A015910, A066606, A062173, A106262.

[Modexp(2,n,n+2): n in [0..120]]; // G. C. Greubel, Jan 11 2023

Table[PowerMod[2, n, n+2], {n, 0, 100}] (* T. D. Noe, Jun 26 2012 *)

print([2**n % (n+2) for n in range(222)])

[power_mod(2,n,n+2) for n in range(121)] # G. C. Greubel, Jan 11 2023

a(n) = 2^n mod (n+2).

a(n) = A106262(2*n, n). - G. C. Greubel, Jan 11 2023

A066607 a(n) = 3^n mod n^3.

Original entry on oeis.org

0, 1, 0, 17, 118, 81, 129, 417, 0, 49, 124, 945, 1498, 177, 1782, 1857, 1958, 729, 3917, 401, 7398, 8721, 6466, 4833, 68, 14049, 0, 3889, 9718, 26649, 24648, 16001, 3294, 6537, 32457, 41553, 34857, 33601, 26703, 32801, 9310, 62721, 76070, 50417, 12393

Offset: 1

Amarnath Murthy, Dec 22 2001

			a(7) = 129 as 3^7 = 2187 = (7^3)*6 + 129.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A066606.

Mathematica
```
Table[ Mod[ 3^n, n^3], {n, 1, 50} ]
```

a(n) = { lift(Mod(3, n^3)^n) } \\ Harry J. Smith, Mar 10 2010

[power_mod(3,n,n^3)for n in range(1,46)] # Zerinvary Lajos, Nov 28 2009

More terms from Floor van Lamoen and Robert G. Wilson v, Dec 23 2001

A233442 a(n) = 2^n mod n^3.

Original entry on oeis.org

0, 4, 8, 16, 32, 64, 128, 256, 512, 24, 717, 640, 1601, 2664, 2393, 0, 3334, 5536, 3004, 576, 4166, 9640, 5545, 8704, 7557, 3696, 19034, 6400, 20244, 5824, 29204, 0, 21293, 12176, 13493, 40960, 41146, 25008, 16154, 11776, 55680, 56008, 20642, 79040, 14957, 1016

Offset: 1

Alex Ratushnyak, Dec 09 2013

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000079, A000578, A066606, A066607.

Cf. A233441, A233471.

Array[PowerMod[2, #, #^3] &, 50] (* Paolo Xausa, Mar 27 2025 *)

for n in range(1,100):  print(2**n % n**3, end=', ')

a(n) = A000079(n) mod A000578(n).

A233471 a(n) = 3^n mod n^2.

Original entry on oeis.org

0, 1, 0, 1, 18, 9, 31, 33, 0, 49, 3, 81, 146, 177, 207, 65, 224, 81, 307, 1, 342, 9, 118, 225, 68, 529, 0, 753, 467, 549, 623, 641, 27, 757, 607, 81, 632, 389, 846, 801, 905, 981, 261, 81, 243, 1757, 1554, 2241, 2383, 249, 792, 1329, 851, 729, 1332, 2529, 1737, 2793, 298

Offset: 1

Alex Ratushnyak, Dec 11 2013

			a(4) = 1 because 3^4 = 81, 4^2 = 16 and 81 = 1 mod 80.
a(5) = 18 because 3^5 = 243, 5^2 = 25 and 243 = 18 mod 25.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000244, A000290, A066606, A066607.

Cf. A233442, A230664.

a:= n-> 3&^n mod n^2:
seq(a(n), n=1..60);  # Alois P. Heinz, Dec 22 2013

Table[Mod[3^n, n^2], {n, 100}] (* Alonso del Arte, Dec 11 2013 *)
Table[PowerMod[3,n,n^2],{n,100}] (* Harvey P. Dale, Aug 27 2019 *)

a(n) = lift(Mod(3, n^2)^n); \\ Michel Marcus, May 05 2021

for n in range(1,100):  print(str(3**n % n**2), end=',')

a(n) = A000244(n) mod A000290(n).

A289382 a(n) = 2^n mod triangular(n).

Original entry on oeis.org

0, 1, 2, 6, 2, 1, 16, 4, 17, 34, 2, 40, 2, 4, 8, 120, 104, 1, 78, 46, 134, 70, 140, 16, 132, 121, 134, 30, 2, 94, 64, 400, 206, 429, 158, 334, 2, 4, 8, 616, 494, 1, 690, 346, 692, 142, 848, 64, 912, 1024, 8, 796, 797, 379, 1528, 4, 350, 178, 1418, 916, 2, 4, 512, 1056, 32, 2011

Offset: 1

Alex Ratushnyak, Jul 04 2017

nonn

			a(6) = 2**6 mod (6*7/2) = 64 mod 21 = 1.

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

Cf. A000079, A000217, A066606, A272934.

seq(2^n mod n*(n+1)/2, n=1..100); # Robert Israel, Jul 04 2017

Table[Mod[2^n, n (n + 1)/2], {n, 66}] (* Michael De Vlieger, Jul 04 2017 *)
PowerMod[2,#,(#(#+1))/2]&/@Range[70] (* Harvey P. Dale, Oct 12 2018 *)

a(n) = lift(Mod(2, n*(n+1)/2)^n); \\ Michel Marcus, Jul 04 2017

for n in range(1,99): print(str(int(2**n % (n*(n+1)/2))), end=', ')

a(n) = A000079(n) mod A000217(n) = 2^n mod n*(n+1)/2.

a(n) = 1 for n>1 in A272934. - Michel Marcus, Jul 04 2017

A066429 a(n) = 7^n mod n^7.

Original entry on oeis.org

0, 49, 343, 2401, 16807, 117649, 0, 1570497, 2089855, 2475249, 9122472, 10209313, 5300159, 98001617, 62916193, 84245889, 82474028, 145795249, 530403874, 57612001, 1611673651, 2006506001, 832663280, 2652148801, 4541072682, 7457168625, 3383428402, 6778582433, 4637847366

Offset: 1

Robert G. Wilson v, Dec 27 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A066606.

Table[ Mod[ 7^n, n^7], {n, 1, 30} ]
Table[PowerMod[7,n,n^7],{n,30}] (* Harvey P. Dale, Dec 25 2023 *)

a(n) = { lift(Mod(7, n^7)^n) } \\ Harry J. Smith, Feb 13 2010

A066430 a(n) = 8^n mod n^8.

Original entry on oeis.org

0, 64, 512, 4096, 32768, 262144, 2097152, 0, 5077565, 73741824, 15579352, 352387072, 769038655, 195124224, 1009588832, 0, 4384458125, 2836131328, 9009163584, 7006846976, 33653509289, 41376995328, 17808619293, 26138902528, 11605068943, 200037316608, 161669290688

Offset: 1

Robert G. Wilson v, Dec 27 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A066606.

Mathematica
```
Table[ Mod[ 8^n, n^8], {n, 1, 30} ]
```

a(n) = { lift(Mod(8, n^8)^n) } \\ Harry J. Smith, Feb 13 2010

[power_mod(8,n,n^8)for n in range(1,24)] # Zerinvary Lajos, Nov 28 2009

A066431 a(n) = 9^n mod n^9.

Original entry on oeis.org

0, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 0, 486784401, 727739626, 3801397473, 7390478182, 5013665073, 26942641524, 68218142337, 50040016962, 127461340881, 20157955560, 179056928801, 95734214766, 798384194161, 812529625653, 123794634177, 1880664942124

Offset: 1

Robert G. Wilson v, Dec 27 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A066606.

Table[ Mod[ 9^n, n^9], {n, 1, 30} ]
Table[PowerMod[9,n,n^9],{n,30}] (* Harvey P. Dale, May 30 2022 *)

a(n) = { lift(Mod(9, n^9)^n) } \\ Harry J. Smith, Feb 13 2010

A066432 a(n) = 10^n mod n^10.

Original entry on oeis.org

0, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 0, 22187726197, 9322172416, 74188586872, 207144033280, 88222656250, 1041257005056, 654556028253, 1391503283200, 4084075030363, 0, 2770357806334, 15498434643968, 29882775483895, 5484153143296

Offset: 1

Robert G. Wilson v, Dec 27 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A066606.

Mathematica
```
Table[ Mod[ 10^n, n^10], {n, 1, 25} ]
```

a(n) = { lift(Mod(10, n^10)^n) } \\ Harry J. Smith, Feb 13 2010

A066444 a(n) = 11^n mod n^11.

Original entry on oeis.org

0, 121, 1331, 14641, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 0, 166394893969, 471664657228, 3140272804489, 8065845196901, 16532101601985, 5101754322287, 57157537871161, 73663224996225, 17332560009201, 240252754633991, 96720656337129, 29113945750083

Offset: 1

Robert G. Wilson v, Dec 27 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A066606.

Mathematica
```
Table[ Mod[11^n, n^11], {n, 1, 20} ]
```

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

cp's OEIS Frontend

A213859 a(n) = 2^n mod (n+2).

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A066607 a(n) = 3^n mod n^3.

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A233442 a(n) = 2^n mod n^3.

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A233471 a(n) = 3^n mod n^2.

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A289382 a(n) = 2^n mod triangular(n).

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A066429 a(n) = 7^n mod n^7.

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A066430 a(n) = 8^n mod n^8.

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