A138649 -id:A138649 - cp's OEIS frontend

A002380 a(n) = 3^n reduced modulo 2^n.

0, 1, 1, 3, 1, 19, 25, 11, 161, 227, 681, 1019, 3057, 5075, 15225, 29291, 55105, 34243, 233801, 439259, 269201, 1856179, 3471385, 6219851, 1882337, 5647011, 50495465, 17268667, 186023729, 21200275, 63600825, 1264544299, 3793632897, 7085931395

Offset: 0

N. J. A. Sloane

A065554 lists the indices n such that a(n+1) = 3*a(n). - Benoit Cloitre, Apr 21 2003

a(n) = (fractional part of (3/2)^n without the decimal point)/5^n = A204544(n) / 5^n. - Michel Lagneau, Jan 25 2012

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 82.
S. S. Pillai, On Waring's problem, J. Indian Math. Soc., 2 (1936), 16-44.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..3322 (first 101 terms from Zak Seidov)
Eric Weisstein's World of Mathematics, Fractional Part.
Eric Weisstein's World of Mathematics, Power Fractional Parts.

Cf. A060692, A002379, A000079, A000244.

Cf. k^n mod (k-1)^n: this sequence (k=3), A064629 (k=4), A138589 (k=5), A138649 (k=6), A139786 (k=7), A138973 (k=8), A139733 (k=9).

a002380 n = 3^n `mod` 2^n  -- Reinhard Zumkeller, Jul 11 2014

a:=n->3^n mod(2^n): seq(a(n), n=0..33); # Zerinvary Lajos, Feb 15 2008

Table[ PowerMod[3, n, 2^n], {n, 0, 33}] (* Robert G. Wilson v, Dec 14 2006 *)
Table[ 3^n - 2^n * Floor[ (3/2)^n ], {n,0,33} ] (* Fred Daniel Kline, Oct 12 2017 *)
x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi;
y[n_] := 3^n - 2^n * x[n];
Array[y, 33] (* Fred Daniel Kline, Dec 21 2017 *)

concat([0],vector(55,n,lift(Mod(3,2^n)^n))) \\ Joerg Arndt, Oct 14 2017

More terms from Jason Earls, Jul 29 2001

A064629 a(n) = 4^n mod 3^n.

Original entry on oeis.org

0, 1, 7, 10, 13, 52, 451, 1075, 6487, 6265, 44743, 119923, 302545, 147298, 589192, 11922706, 33341917, 4227505, 146050183, 584200732, 1174541461, 4698165844, 18792663376, 43789593895, 175158375580, 700633502320, 1955245399837, 2737249942690, 18574597255747

Offset: 0

Labos Elemer, Oct 01 2001

(a(n+1) - 4*a(n))/3^n is always one of -3, -2, -1, 0, 1, 2. - Robert Israel, Dec 01 2016

Harry J. Smith and Seiichi Manyama, Table of n, a(n) for n = 0..2096 (first 201 terms from Harry J. Smith)

Cf. A002379, A060692, A064628.

Cf. k^n mod (k-1)^n: A002380 (k=3), this sequence (k=4), A138589 (k=5), A138649 (k=6), A139786 (k=7), A138973 (k=8), A139733 (k=9).

[4^n mod 3^n: n in [0..40]]; // Wesley Ivan Hurt, Dec 01 2016

A064629:=n->4^n mod 3^n: seq(A064629(n), n=0..40); # Wesley Ivan Hurt, Dec 01 2016

Table[PowerMod[4,n,3^n],{n,0,30}] (* Harvey P. Dale, Aug 26 2012 *)

{ f=t=1; for (n=0, 200, write("b064629.txt", n, " ", f%t); f*=4; t*=3 ) } \\ Harry J. Smith, Sep 20 2009

[power_mod(4,n,3^n)for n in range(0,27)] # Zerinvary Lajos, Nov 28 2009

a(26) from Harry J. Smith, Sep 20 2009

A138589 a(n) = 5^n mod 4^n.

Original entry on oeis.org

0, 1, 9, 61, 113, 53, 3337, 12589, 62945, 118117, 328441, 2690781, 9259601, 12743573, 197935593, 452807053, 2264035265, 7025209029, 35126045145, 106910748989, 809431651889, 1848135003893, 9240675019465, 28611189052909, 213424689442209, 785648470500389

Offset: 0

N. J. A. Sloane, May 20 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1661

Cf. k^n mod (k-1)^n: A002380 (k=3), A064629 (k=4), this sequence (k=5), A138649 (k=6), A139786 (k=7), A138973 (k=8), A139733 (k=9).

Table[PowerMod[5, n, 4^n], {n,0,50}] (* G. C. Greubel, Oct 01 2017 *)

concat([0], vector(50, n, lift(Mod(5, 4^n)^n))) \\ Michel Marcus, Oct 02 2017

A138973 a(n) = 8^n mod 7^n.

Original entry on oeis.org

0, 1, 15, 169, 1695, 15961, 26846, 450066, 5247614, 13156907, 226316077, 680627620, 13354327932, 65310761853, 328708074010, 1951441519231, 15611532153848, 158125187800385, 101848932467045, 7328445851378156, 35829776440278962, 286638211522231696

Offset: 0

N. J. A. Sloane, May 20 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1183

Cf. k^n mod (k-1)^n: A002380 (k=3), A064629 (k=4), A138589 (k=5), A138649 (k=6), A139786 (k=7), this sequence (k=8), A139733 (k=9).

Cf. A000420, A001018.

a[n_]:=PowerMod[8,n,7^n];Array[a,22,0] (* James C. McMahon, Jun 23 2025 *)

a(n) = lift(Mod(8, 7^n)^n); \\ Michel Marcus, Feb 20 2018

[power_mod(8,n,7^n) for n in range(0,22)] # Zerinvary Lajos, Nov 28 2009

A139786 a(n) = 7^n mod 6^n.

Original entry on oeis.org

0, 1, 13, 127, 1105, 1255, 24337, 263671, 725953, 42823, 40610545, 163341463, 780593185, 5464152295, 51309760081, 45711664183, 2200721587585, 12583941205639, 3454291215793, 430419865184215, 3012939056289505, 10122098073837607, 92791637157241105, 517919756258420599

Offset: 0

N. J. A. Sloane, May 20 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1285

Cf. k^n mod (k-1)^n: A002380 (k=3), A064629 (k=4), A138589 (k=5), A138649 (k=6), this sequence (k=7), A138973 (k=8), A139733 (k=9).

A245805 a(n) = 12^n mod 11^n.

Original entry on oeis.org

0, 1, 23, 397, 6095, 87781, 1214423, 16344637, 1263934, 443884970, 10042515022, 172385029466, 2639243694814, 3425068947279, 144668963799141, 2875277066339415, 1085339440747772, 196822992743261908, 4383664026916317980, 13684547128550195393, 470010017784675076171

Offset: 0

Vincenzo Librandi, Aug 04 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..970

Cf. k^n mod (k-1)^n: A002380 (k=3), A064629 (k=4), A138589 (k=5), A138649 (k=6), A139786 (k=7), A138973 (k=8), A139733 (k=9), A016189 (k=10), A016195 (k=11), this sequence (k=12),

Magma
```
[12^n mod 11^n: n in [0..25]];
```

Table[PowerMod[12, n, 11^n], {n, 0, 30}]

vector(50, n, 12^(n-1)%11^(n-1)) \\ Derek Orr, Aug 04 2014

cp's OEIS Frontend

A002380 a(n) = 3^n reduced modulo 2^n.

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A064629 a(n) = 4^n mod 3^n.

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A138589 a(n) = 5^n mod 4^n.

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A138973 a(n) = 8^n mod 7^n.

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A139786 a(n) = 7^n mod 6^n.

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A245805 a(n) = 12^n mod 11^n.

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