A004656 -id:A004656 - cp's OEIS frontend

A070972 Length of longest run of consecutive 1's in binary expansion of 3^n (A004656).

1, 2, 1, 2, 1, 4, 2, 2, 2, 3, 3, 7, 6, 4, 5, 4, 3, 4, 4, 4, 6, 5, 5, 4, 3, 2, 10, 8, 7, 5, 4, 5, 7, 9, 8, 6, 5, 7, 11, 10, 8, 6, 6, 4, 9, 7, 6, 4, 9, 8, 6, 9, 8, 6, 5, 5, 3, 4, 7, 5, 10, 8, 7, 6, 6, 6, 6, 4, 4, 7, 7, 5, 5, 5, 5, 5, 5, 9, 8, 6, 5, 6, 5, 6, 5, 5, 4, 6, 5, 10, 8, 7, 5, 6, 6, 6, 7, 7, 6, 7, 5, 5, 5, 9

Offset: 0

Frank Schwellinger (nummer_eins(AT)web.de), Jan 03 2003

			3^11 = (101011001111111011) binary, so a(11) = 7.

Cf. A004656 and A011754.

f[n_Integer] := Block[{p = Flatten[ Position[ Prepend[ IntegerDigits[2*3^n, 2], 0], 0]]}, Max[Drop[p, 1] - Drop[p, -1]] - 1]; Table[ f[n], {n, 0, 103}]

Edited by Robert G. Wilson v, Jan 04 2002

A004642 Powers of 2 written in base 3.

Original entry on oeis.org

1, 2, 11, 22, 121, 1012, 2101, 11202, 100111, 200222, 1101221, 2210212, 12121201, 102020102, 211110211, 1122221122, 10022220021, 20122210112, 111022121001, 222122012002, 1222021101011, 10221112202022, 21220002111121, 120210012000012, 1011120101000101, 2100010202000202

Offset: 0

N. J. A. Sloane

When n is odd, a(n) ends in 1, and when n is even, a(n) ends in 2, since 2^n is congruent to 1 mod 3 when n is odd and to 2 mod 3 when n is even. - Alonso del Arte Dec 11 2009
Sloane (1973) conjectured a(n) always has a 0 between the most and least significant digits if n > 15 (see A102483 and A346497).
Erdős (1978) conjectured that for n > 8 a(n) has at least one 2 (see link to Terry Tao's blog). - Dmitry Kamenetsky, Jan 10 2017

N. J. A. Sloane, The Persistence of a Number, J. Recr. Math. 6 (1973), 97-98.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Yagub N. Aliyev, Digits of powers of 2 in ternary numeral system, Notes on Number Theory and Discrete Mathematics, Vol. 29, No. 3 (2023), 474-485.
Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 67-70.
Donald L. Kreher and Douglas R. Stinson, On min-base palindromic representations of powers of 2, arXiv:2401.07351 [math.NT], 2024. See Table 4 p. 10.
Jeffrey C. Lagarias, Ternary Expansions of Powers of 2, Journal of the London Mathematical Society, Vol. 79, No. 3 (2009), pp. 562-588; arXiv preprint, arXiv:math/0512006 [math.DS], 2005-2008.
Terry Tao, The Collatz Conjecture, Littlewood-Offord theory, and powers of 2 and 3, 2011.
Eric Weisstein's World of Mathematics, Ternary.

Cf. A000079: powers of 2 written in base 10.
Cf. A004643, ..., A004655: powers of 2 written in base 4, 5, ..., 16.
Cf. A004656, A004658, A004659, ..., A004663: powers of 3 written in base 2, 4, 5, ..., 9.

[Seqint(Intseq(2^n, 3)): n in [0..30]]; // G. C. Greubel, Sep 10 2018

Table[FromDigits[IntegerDigits[2^n, 3]], {n, 25}] (* Alonso del Arte Dec 11 2009 *)

a(n)=fromdigits(digits(2^n,3)) \\ M. F. Hasler, Jun 23 2018

A011754 Number of ones in the binary expansion of 3^n.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 6, 5, 6, 8, 9, 13, 10, 11, 14, 15, 11, 14, 14, 17, 17, 20, 19, 22, 16, 18, 24, 30, 25, 25, 25, 26, 26, 34, 29, 32, 27, 34, 36, 32, 28, 39, 38, 39, 34, 34, 45, 38, 41, 33, 41, 46, 42, 35, 39, 42, 39, 40, 42, 48, 56, 56, 49, 57, 56, 51, 45, 47, 55, 55, 64, 68, 58

Offset: 0

Allan C. Wechsler, Dec 11 1999

Conjecture: a(n)/n tends to log(3)/(2*log(2)) = 0.792481250... (A094148). - Ed Pegg Jr, Dec 05 2002
Senge & Straus prove that for every m, there is some N such that for all n > N, a(n) > m. Dimitrov & Howe make this effective, proving that for n > 25, a(n) > 22. - Charles R Greathouse IV, Aug 23 2021
Ed Pegg's conjecture means that about half of the bits of 3^n are nonzero. It appears that the same is true for 5^n (A000351, cf. A118738) and 7^n (A000420). - M. F. Hasler, Apr 17 2024

S. Wolfram, "A new kind of science", p. 903.

Hugo Pfoertner, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Vassil S. Dimitrov and Everett W. Howe, Powers of 3 with few nonzero bits and a conjecture of Erdős, arXiv:2105.06440 [math.NT], 2021.
Taylor Dupuy and David E. Weirich, Bits of 3^n in binary, Wieferich primes and a conjecture of Erdős, Journal of Number Theory, Volume 158, January 2016, Pages 268-280.
Hugo Pfoertner, Plot of a(n) - 0.79248*n, +-Pi*sqrt(n), n up to 10^6.
H. G. Senge and E. G. Straus, PV-numbers and sets of multiplicity, Periodica Mathematica Hungarica 3 (1973), pp. 93-100.

Cf. A007088, A000120 (Hamming weight), A000244 (3^n), A004656, A261009, A094148.

Cf. A118738 (same for 5^n).

a011754 = a000120 . a000244  -- Reinhard Zumkeller, Aug 14 2015

[&+Intseq(3^n, 2): n in [0..79]]; // Vincenzo Librandi, Nov 28 2018

f:= n -> convert(convert(3^n,base,2),`+`):
map(f, [$0..100]); # Robert Israel, Apr 17 2024

Table[DigitCount[3^n, 2][[1]], {n, 0, 100}] (* Stefan Steinerberger, Apr 03 2006 *)
DigitCount[3^Range[0,100],2,1] (* Harvey P. Dale, Apr 06 2012 *)

a(n)=hammingweight(3^n) \\ Charles R Greathouse IV, Feb 09 2017

A011754 = lambda n: (3**n).bit_count() # M. F. Hasler, Apr 17 2024

a(n) = A000120(3^n). - Benoit Cloitre, Dec 06 2002

a(n) = A000120(A000244(n)). - Reinhard Zumkeller, Aug 14 2015

More terms from Stefan Steinerberger, Apr 03 2006

A004655 Powers of 2 written in base 16.

Original entry on oeis.org

1, 2, 4, 8, 10, 20, 40, 80, 100, 200, 400, 800, 1000, 2000, 4000, 8000, 10000, 20000, 40000, 80000, 100000, 200000, 400000, 800000, 1000000, 2000000, 4000000, 8000000, 10000000, 20000000, 40000000, 80000000

Offset: 0

N. J. A. Sloane

10^(Floor[n/4]) | a(n). The first term of each value cycles the pattern {1, 2, 4, 8}. - G. C. Greubel, Sep 10 2018

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,0,0,10).

Cf. A000079, A004643, ..., A004654: powers of 2 written in base 10, 4, 5, ..., 15.

Cf. A000244, A004656, A004658, A004659, ... : powers of 3 in base 10, 2, 4, 5, ...

[Seqint(Intseq(2^n, 16)): n in [0..30]]; // G. C. Greubel, Sep 10 2018

Table[FromDigits[IntegerDigits[2^n, 16]], {n, 50}] (* G. C. Greubel, Sep 11 2018 *)

apply( a(n)=2^(n%4)*10^(n\4), [0..30]) \\ M. F. Hasler, Jun 22 2018

def A004655(n): return 10**(n>>2)<<(n&3) # Chai Wah Wu, Jan 27 2023

a(n) = 2^(n mod 4)*10^floor(n/4). - M. F. Hasler, Jun 22 2018
From Chai Wah Wu, Sep 03 2020: (Start)
a(n) = 10*a(n-4) for n > 3.
G.f.: -(2*x + 1)*(4*x^2 + 1)/(10*x^4 - 1). (End)

A004658 Powers of 3 written in base 4.

Original entry on oeis.org

1, 3, 21, 123, 1101, 3303, 23121, 202023, 1212201, 10303203, 32122221, 223033323, 2001233301, 12011033103, 102033231321, 312233021223, 2210031131001, 13230220113003, 113011321011021, 1011101223033123

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000244, A004656, A004659, ... : powers of 3 written in base 10, 2, 5, ...

Cf. A000079, A004643, ..., A004655: powers of 2 written in base 10, 4, 5, ..., 16

[Seqint(Intseq(3^n, 4)): n in [0..30]]; // G. C. Greubel, Sep 10 2018

Table[FromDigits[IntegerDigits[3^n, 4]], {n, 0, 40}] (* Vincenzo Librandi, Jun 07 2013 *)

a(n,b=4,m=3)=fromdigits(digits(m^n,b)) \\ M. F. Hasler, Jun 22 2018

A000866 2^n written in base 5.

Original entry on oeis.org

1, 2, 4, 13, 31, 112, 224, 1003, 2011, 4022, 13044, 31143, 112341, 230232, 1011014, 2022033, 4044121, 13143242, 31342034, 113234123, 232023301, 1014102102, 2033204204, 4121413413, 13243332331, 32042220212, 114134440424, 233324431403, 1022204413311

Offset: 0

N. J. A. Sloane, Jacques Haubrich (jhaubrich(AT)freeler.nl)

T. D. Noe, Table of n, a(n) for n = 0..300

Cf. A000079, A004642, ..., A004655: powers of 2 written in base 10, 3, 4, ..., 16

Cf. A000244, A004656, A004658, A004659, ... : powers of 3 written in base 10, 2, 4, 5, ...

Table[FromDigits[IntegerDigits[2^n, 5]], {n, 0, 30}] (* T. D. Noe, Jun 20 2012 *)

a(n)=fromdigits(digits(2^n,5)) \\ M. F. Hasler, Jun 23 2018

More terms from Erich Friedman.

A037095 "Sloping binary representation" of powers of 3 (A000244), slope = -1.

Original entry on oeis.org

1, 1, 3, 1, 3, 9, 11, 17, 19, 25, 123, 65, 195, 169, 171, 753, 435, 249, 2267, 4065, 8163, 841, 843, 31313, 29651, 39769, 38331, 30081, 160643, 49769, 53867, 563377, 700659, 1611961, 760731, 1207073, 5668771, 5566345, 11844619, 8699025, 10386067, 55868313

Offset: 0

Antti Karttunen, Jan 28 1999

			When powers of 3 are written in binary (see A004656), under each other as:
  000000000001 (1)
  000000000011 (3)
  000000001001 (9)
  000000011011 (27)
  000001010001 (81)
  000011110011 (243)
  001011011001 (729)
  100010001011 (2187)
and one collects their bits from the column-0 to NW-direction (from the least to the most significant end), one gets 1 (1), 01 (1), 011 (3), 0001 (1), 00011 (3), 001001 (9), etc. (See A105033 for similar transformation done on nonnegative integers, A001477).

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A105033, A000244, A037093, A037094, A037096, A037097, A339601.

A037095:= n-> add(bit_n(3^(n-i), i)*(2^i), i=0..n):
bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2):
seq(A037095(n), n=0..41);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, (p->
       expand((p-(p mod 2))*x/2)+3^n)(b(n-1)))
    end:
a:= n-> subs(x=2, b(n) mod 2):
seq(a(n), n=0..42);  # Alois P. Heinz, Dec 10 2020

A339601(n) = { my(m=1, s=0); while(n>=m, s += bitand(m,n); m <<= 1; n \= 3); (s); };
A037095(n) = A339601(3^n); \\ Antti Karttunen, Dec 09 2020

BINSLOPE(f) = n -> sum(i=0,n,bitand(2^(n-i),f(i))); \\ General transformation for these kinds of sequences.
A037095 = BINSLOPE(n -> 3^n); \\ And its application to A000244. - Antti Karttunen, Dec 09 2020

a(n) = A339601(A000244(n)). - Antti Karttunen, Dec 09 2020

Entry revised Dec 29 2007

More terms from Sean A. Irvine, Dec 08 2020

A037096 Periodic vertical binary vectors computed for powers of 3: a(n) = Sum_{k=0 .. (2^n)-1} (floor((3^k)/(2^n)) mod 2) * 2^k.

Original entry on oeis.org

1, 2, 0, 204, 30840, 3743473440, 400814250895866480, 192435610587299441243182587501623263200, 2911899996313975217187797869354128351340558818020188112521784134070351919360

Offset: 0

Antti Karttunen, Jan 29 1999

This sequence can be also computed with a recurrence that does not explicitly refer to 3^n. See the C program.

Conjecture: For n >= 3, each term a(n), when considered as a GF(2)[X] polynomial, is divisible by the GF(2)[X] polynomial (x + 1) ^ A055010(n-1). If this holds, then for n >= 3, a(n) = A048720(A136386(n), A048723(3,A055010(n-1))).

			When powers of 3 are written in binary (see A004656), under each other as:
  000000000001 (1)
  000000000011 (3)
  000000001001 (9)
  000000011011 (27)
  000001010001 (81)
  000011110011 (243)
  001011011001 (729)
  100010001011 (2187)
it can be seen that the bits in the n-th column from the right can be arranged in periods of 2^n: 1, 2, 4, 8, ... This sequence is formed from those bits: 1, is binary for 1, thus a(0) = 1. 01, reversed is 10, which is binary for 2, thus a(1) = 2, 0000 is binary for 0, thus a(2)=0, 000110011, reversed is 11001100 = A007088(204), thus a(3) = 204.

S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

Antti Karttunen, Table of n, a(n) for n = 0..11
Antti Karttunen, C program for computing this sequence.
S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

Cf. A036284, A037095, A037097, A136386 for related sequences.
Cf. A000079, A000215, A000225, A000244, A004656, A007088, A048720, A048723, A055010.
Cf. also A004642, A265209, A265210 (for 2^n written in base 3).

a(n) := sum( 'bit_n(3^i, n)*(2^i)', 'i'=0..(2^(n))-1);
bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2);

a(n) = Sum_{k=0 .. A000225(n)} (floor(A000244(k)/(2^n)) mod 2) * 2^k.
Other identities and observations:
For n >= 2, a(n) = A000215(n-1)*A037097(n) = A048720(A037097(n), A048723(3, A000079(n-1))).

Entry revised by Antti Karttunen, Dec 29 2007

Name changed and the example corrected by Antti Karttunen, Dec 05 2015

A004668 Powers of 3 written in base 26. (Next term contains a non-decimal digit.)

Original entry on oeis.org

1, 3, 9, 11, 33, 99, 121, 363

Offset: 0

N. J. A. Sloane, Dec 11 1996

Aliquot divisors of 1089. - Omar E. Pol, Jun 10 2014

The above comment refers to the first 8 terms only. The next term would contain a digit 18, commonly coded as I, if A, B, ... are used for digits > 9. But this does not mean that the sequence is finite. Many other encodings of digits > 9 are conceivable (e.g., using 000, 100, 110, ..., 250 for digits 0, 10, 11, ..., 25). - M. F. Hasler, Jun 22 2018

Cf. A000244, A004656, A004658, A004659, ..., A004667: powers of 3 in base 10, 2, 4, 5, ..., 13.

Cf. A000079, A004643, ..., A004655: powers of 2 written in base 10, 4, 5, ..., 16.

Select[Divisors[1089], # < 1089 &] (* Wesley Ivan Hurt, Jun 13 2014 *)

fordiv(1089, d, (d<1089) && print1(d, ", ")) \\ Michel Marcus, Jun 14 2014

divisors(1089)[^-1] \\ M. F. Hasler, Jun 22 2018

apply( A004668(n,b=26,m=3)=fromdigits(digits(m^n,b)), [0..8]) \\ This implements one possible continuation of the sequence beyond n = 7: write digits in decimal and carry over (so 363*3 = 9I9[26] -> 9*100 + 18*10 + 9 = 1089). - M. F. Hasler, Jun 22 2018

A078839 Numbers k such that the binary expansion of 3^k has the same number of 0's and 1's.

Original entry on oeis.org

2, 12, 69, 73, 150, 184, 252, 328, 339, 464, 483, 541, 729, 747, 758, 763, 1014, 1047, 1090, 1094, 1158, 1264, 1359, 1601, 1679, 1693, 1698, 1780, 2368, 2641, 2815, 3292, 3393, 3606, 3682, 3857, 3909, 3919, 3963, 4087, 4111, 4289, 4314, 5017, 5398, 5466

Offset: 1

Benoit Cloitre, Dec 06 2002

Does the limit of a(n)/n^2 as n -> infinity exist?

Hugo Pfoertner, Table of n, a(n) for n = 1..1600 (terms 1..1000 from Amiram Eldar)
Amiram Eldar, Plot of a(n)/n^2 for n = 1..1000
Hugo Pfoertner, Plot of a(n)/n^2 using Plot 2.

Cf. A000244, A004656, A011754, A031443.

balanced[n_] := Module[{d=IntegerDigits[n, 2]}, Plus@@d==Length[d]/2]; Select[Range[0, 5500], balanced[3^# ]&]

is(n)=hammingweight(n=3^n)==hammingweight(bitneg(n,#binary(n))) \\ Charles R Greathouse IV, Mar 29 2013

cp's OEIS Frontend

A070972 Length of longest run of consecutive 1's in binary expansion of 3^n (A004656).

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A004642 Powers of 2 written in base 3.

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A011754 Number of ones in the binary expansion of 3^n.

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A004655 Powers of 2 written in base 16.

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A004658 Powers of 3 written in base 4.

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A000866 2^n written in base 5.

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A037095 "Sloping binary representation" of powers of 3 (A000244), slope = -1.

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