A064629 -id:A064629 - 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

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

A138649 a(n) = 6^n mod 5^n.

Original entry on oeis.org

0, 1, 11, 91, 46, 1526, 15406, 45561, 117116, 312071, 1872426, 21000181, 223657336, 853662766, 5121976596, 12421312701, 74527876206, 141991475986, 2377827762166, 18081663838621, 32196037719226, 2441363034106, 491485336407761, 2948912018446566

Offset: 0

N. J. A. Sloane, May 20 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1430

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

Table[PowerMod[6,n,5^n],{n,0,30}] (* Harvey P. Dale, Jan 23 2014 *)

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).

A064630 Number of parts if 4^n is partitioned into parts of size 3^n as far as possible into parts of size 2^n as far as possible and into parts of size 1^n.

Original entry on oeis.org

2, 5, 5, 16, 25, 15, 66, 121, 146, 771, 1220, 3641, 8093, 15843, 28359, 50236, 33366, 36709, 145250, 137776, 548024, 2186496, 1066102, 4251976, 16984368, 28678103, 13620614, 205950171, 100716646, 381399635, 1397934923, 3826001641

Offset: 1

Labos Elemer, Oct 01 2001

nonn

Corresponds to the only solution of the Diophantine equation 4^n = x*3^n + y*2^n + z*1^n with constraints 0 <= y < 3^n/2^n, 0 <= z < 2^n.

Binary order (cf. A029837) of a(n) is close to n.

			4^6 = 4096 = 729 + 729 + 729 + 729 + 729 + 64 + 64 + 64 + 64 + 64 + 64 + 64 + 1 + 1 + 1 = 5*3^6 + 7*2^6 + 3*1^6, so a(6) = 5 + 7 + 3 = 15.

Harry J. Smith, Table of n, a(n) for n = 1..250

Cf. A064628, A064629, A060692, A029837.

{for(n=1,32,a=divrem(4^n,3^n); b=divrem(a[2],2^n); print1(a[1]+b[1]+b[2],","))}

{ f=t=w=1; for (n=1, 250, f*=4; t*=3; w*=2; a=divrem(f, t); b=divrem(a[2], w); write("b064630.txt", n, " ", a[1]+b[1]+b[2]) ) } \\ Harry J. Smith, Sep 20 2009

a(n) = A064628(n) + floor(A064629(n)/2^n) + (A064629(n) mod 2^n) = floor(4^n/3^n) + floor((4^n mod 3^n)/2^n) + ((4^n mod 3^n) mod 2^n)

Edited by Klaus Brockhaus, May 24 2003

A264921 Decimal expansion of constant z = Sum_{n>=1} {(4/3)^n} * (3/4)^n, where {x} is the fractional part of x.

Original entry on oeis.org

1, 3, 6, 7, 4, 6, 1, 3, 7, 9, 3, 5, 3, 3, 2, 9, 2, 6, 9, 0, 2, 1, 3, 0, 0, 5, 2, 8, 2, 3, 7, 5, 4, 0, 8, 0, 4, 3, 4, 5, 9, 4, 5, 5, 1, 2, 8, 4, 8, 9, 9, 5, 3, 0, 8, 3, 7, 2, 0, 4, 7, 8, 1, 1, 2, 5, 6, 7, 4, 0, 4, 6, 8, 0, 2, 1, 0, 7, 3, 8, 6, 8, 3, 6, 3, 9, 2, 4, 7, 1, 7, 6, 6, 7, 7, 1, 9, 8, 5, 1, 0, 6, 6, 5, 7, 1, 2, 6, 3, 8, 2, 0, 9, 1, 4, 3, 0, 0, 9, 3, 2, 6, 2, 8, 0, 9, 3, 8, 9, 8, 7, 7, 0, 2, 2, 9, 6, 1, 1, 0, 1, 6, 8, 2, 1, 7, 2, 4, 9, 9, 0, 2, 2, 3, 8, 2, 5, 9, 3, 4, 1, 8, 1, 6, 5, 5, 4, 5, 9, 5, 0, 0, 8, 5, 3, 6, 4, 1, 9, 1, 0, 5, 7, 2, 4, 4, 3, 2, 9

Offset: 1

Paul D. Hanna, Dec 03 2015

			z = 1.3674613793533292690213005282375408043459455128489\
95308372047811256740468021073868363924717667719851\
06657126382091430093262809389877022961101682172499\
02238259341816554595008536419105724432961711520592\
92511101423029805093364719414748469451590148076361\
52981353989027739504422481304813339179550172220838\
78986350689080620566812697277477621308107983782819\
76274774500215875970544025343446657398435575812229\
28979675592867430344641751297842513480112243120370\
37616509374801184872891959991759744341259271254468...
INFINITE SERIES.
z = 1/4 + 7/4^2 + 10/4^3 + 13/4^4 + 52/4^5 + 451/4^6 + 1075/4^7 + 6487/4^8 + 6265/4^9 + 44743/4^10 + 119923/4^11 + 302545/4^12 + 147298/4^13 + 589192/4^14 + 11922706/4^15 + 33341917/4^16 + 4227505/4^17 + 146050183/4^18 + 584200732/4^19 + 1174541461/4^20 +...+ A064629(n)/4^n +...

Cf. A064629 (4^n mod 3^n), A264918, A264919, A264920, A264922.

z = Sum_{n>=1} (4^n mod 3^n) / 4^n = Sum_{n>=1} A064629(n) / 4^n.

A064631 a(n) = ceiling(log_2(A064630(n))).

Original entry on oeis.org

2, 3, 3, 5, 5, 4, 7, 7, 8, 10, 11, 12, 13, 14, 15, 16, 16, 16, 18, 18, 20, 22, 21, 23, 25, 25, 24, 28, 27, 29, 31, 32, 33, 34, 35, 36, 37, 37, 39, 40, 39, 42, 42, 44, 44, 46, 46, 46, 49, 50, 51, 51, 51, 54, 55, 55, 57, 57, 59, 60, 60, 61, 63, 64, 64, 66, 60, 62, 67, 70, 69, 72

Offset: 1

Labos Elemer, Oct 01 2001

nonn

In A064630, using a greedy algorithm we write 4^n = x*3^n+y*2^n+z*1^n and A064630(n) = x+y+z. This sequence is a measure of the "length" or complexity of those solutions.

Cf. A002379, A002380, A060692, A064628, A064629, A064630, A029837.

a(n) = A029837(A064630(n)) = ceiling(log_2(A064630(n))).

Initial terms corrected and entry revised by Sean A. Irvine, Jul 18 2023

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

A178998 Primes of the form 4^k mod 3^k.

Original entry on oeis.org

7, 13, 119923, 146050183, 4039362385345521139, 289247481259011497824466400997481269, 1765256712749403700417549596608786383, 395766070055468241613007225643003404495980782673, 2596786183076854435238229837938226284218037897451862682304077097493117

Offset: 1

Juri-Stepan Gerasimov, Jan 03 2011

nonn

Cf. A000040, A000079, A000244, A064629, A178985.

select(isprime, [4&^n mod 3^n$n=1..200])[];  # Alois P. Heinz, May 18 2019

Select[Table[PowerMod[4, n, 3^n], {n, 100}], PrimeQ] (* Alonso del Arte, Jan 03 2011 *)

terms(n) = my(i=0); for(k=0, oo, if(i>=n, break); my(x=lift(Mod(4, 3^k)^k)); if(ispseudoprime(x), print1(x, ", "); i++))
/* Print initial 7 terms as follows: */
terms(7) \\ Felix Fröhlich, May 18 2019

{ A000040 } intersect { A064629 }.

cp's OEIS Frontend

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

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

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A138649 a(n) = 6^n mod 5^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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A064630 Number of parts if 4^n is partitioned into parts of size 3^n as far as possible into parts of size 2^n as far as possible and into parts of size 1^n.

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A264921 Decimal expansion of constant z = Sum_{n>=1} {(4/3)^n} * (3/4)^n, where {x} is the fractional part of x.

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A064631 a(n) = ceiling(log_2(A064630(n))).

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

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