A004643 -id:A004643 - cp's OEIS frontend

A004685 Fibonacci numbers written in base 2.

0, 1, 1, 10, 11, 101, 1000, 1101, 10101, 100010, 110111, 1011001, 10010000, 11101001, 101111001, 1001100010, 1111011011, 11000111101, 101000011000, 1000001010101, 1101001101101, 10101011000010, 100010100101111, 110111111110001, 1011010100100000, 10010010100010001

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wayne State University College of Engineering, Engineering Bits & Bytes: The Fibonacci Puzzle, (Video, this sequence is looped through in 16-bit words close to the beginning and again at the end).

Cf. A004686 .. A004694: Fibonacci numbers written in base 3, 4, ..., 13.
Cf. A004676 .. A004684: Primes written in base 2, 3, 4, ..., 11.
Cf. A004643, ..., A004668 : powers of 2 resp. of 3 in base 3, 4, 5, ..., 26.

[Seqint(Intseq(Fibonacci(n),2)): n in [0..50]]; // G. C. Greubel, Oct 09 2018

with(combinat): seq(convert(fibonacci(n),binary),n=0..25); # Muniru A Asiru, Oct 10 2018

Table[FromDigits[IntegerDigits[Fibonacci[n], 2]], {n, 0, 30}] (* Stefan Steinerberger, Apr 14 2006 *)

a(n)=subst(Pol(binary(fibonacci(n))),'x,10) \\ Charles R Greathouse IV, Feb 03 2014

apply( n->fromdigits(binary(fibonacci(n))), [0..19]) \\ M. F. Hasler, Jun 22 2018

vector(50, n, n--; fromdigits(digits(fibonacci(n), 2))) \\ G. C. Greubel, Oct 09 2018

a(n) = A007088(A000045(n)). - Jonathan Vos Post, Aug 24 2010

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

A004656 Powers of 3 written in base 2.

Original entry on oeis.org

1, 11, 1001, 11011, 1010001, 11110011, 1011011001, 100010001011, 1100110100001, 100110011100011, 1110011010101001, 101011001111111011, 10000001101111110001, 110000101001111010011, 10010001111101101111001

Offset: 0

N. J. A. Sloane

S. Wolfram, A New Kind of Science, Wolfram Media, 2002, pp. 120 and 903.

Vincenzo Librandi, Table of n, a(n) for n = 0..640

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

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

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

Table[ FromDigits[ IntegerDigits[3^n, 2]], {n, 0, 14}]

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

A073053 Apply DENEAT operator (or the Sisyphus function) to n.

Original entry on oeis.org

101, 11, 101, 11, 101, 11, 101, 11, 101, 11, 112, 22, 112, 22, 112, 22, 112, 22, 112, 22, 202, 112, 202, 112, 202, 112, 202, 112, 202, 112, 112, 22, 112, 22, 112, 22, 112, 22, 112, 22, 202, 112, 202, 112, 202, 112, 202, 112, 202, 112, 112, 22, 112, 22

Offset: 0

Michael Joseph Halm, Aug 16 2002

DENEAT(n): concatenate number of even digits in n, number of odd digits and total number of digits. E.g., 25 -> 1.1.2 = 112 (Digits: Even, Not Even, And Total). Leading zeros are then omitted.
This is also known as the Sisyphus function. - N. J. A. Sloane, Jun 25 2018
Repeated application of the DENEAT operator reduces all numbers to 123. This is easy to prove. Compare A073054, A100961. - N. J. A. Sloane Jun 18 2005

			a(1) = 0.1.1 -> 11.
a(10000000000) = 10111 because 10000000000 has 10 even digits, 1 odd digit and 11 total digits

M. E. Coppenbarger, Iterations of a modified Sisyphus function, Fib. Q., 56 (No. 2, 2018), 130-141.
M. Ecker, Caution: Black Holes at Work, New Scientist (Dec. 1992)
M. J. Halm, Blackholing, Mpossibilities 69, (Jan 01 1999), p. 2.
J. Schram, The Sisyphus string, J. Rec. Math., 19:1 (1987), 43-44.
M. Zeger, Fatal attraction, Mathematics and Computer Education, 27:2 (1993), 118-123.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A008577, A072420, A073054, A100961, A171797.

For records see A305395, A004643, A308004.

read("transforms") :
A073053 := proc(n)
    local e,o,L ;
    if n = 0 then
        0 ;
    else
        e := A196563(n) ;
        o := A196564(n) ;
        L := [e,o,e+o] ;
        digcatL(L) ;
    end if;
end proc: # R. J. Mathar, Jul 13 2012
# Maple code based on R. J. Mathar's code for A171797, added by N. J. A. Sloane, May 12 2019 (Start)
nevenDgs := proc(n) local a, d; a := 0 ; for d in convert(n, base, 10) do if type(d, 'even') then a :=a +1 ; end if; end do; a ; end proc:
cat2 := proc(a, b) local ndigsb; ndigsb := max(ilog10(b)+1, 1) ; a*10^ndigsb+b ; end:
catL := proc(L) local a, i; a := op(1, L) ; for i from 2 to nops(L) do a := cat2(a, op(i, L)) ; end do; a; end proc:
A055642 := proc(n) max(1, ilog10(n)+1) ; end proc:
A171797 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n1, n2, n1-n2]) ; end proc:
A073053 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n2, n1-n2, n1]) ; end proc:
seq(A073053(n), n=1..80) ; (End)
L:=proc(n) if n=0 then 1 else floor(evalf(log(n)/log(10)))+1; fi; end;
S:=proc(n) local Le,Ld,Lt,t1,e,d,t; global L;
t1:=convert(n,base,10); e:=0; d:=0; t:=nops(t1);
for i from 1 to t do if (t1[i] mod 2) = 0 then e:=e+1; else d:=d+1; fi; od:
Le:=L(e); Ld:=L(d); Lt:=L(t);
if e=0 then 10^Lt*d+t
elif d=0 then 10^(Ld+Lt)*e+10^Lt*d+t
else 10^(Ld+Lt)*e+10^Lt*d+t; fi;
end;
[seq(S(n),n=1..200)]; # N. J. A. Sloane, Jun 25 2018
# alternative Maple program:
a:= n-> (l-> (e-> parse(cat(e, (h-> [h-e, h][])(nops(l))))
    )(nops(select(x-> x::even, l))))(convert(n, base, 10)):
seq(a(n), n=0..200);  # Alois P. Heinz, Jan 21 2022

f[n_] := Block[{id = IntegerDigits[n]}, FromDigits[ Join[ IntegerDigits[ Length[ Select[id, EvenQ[ # ] &]]], IntegerDigits[ Length[ Select[id, OddQ[ # ] &]]], IntegerDigits[ Length[ id]] ]]]; Table[ f[n], {n, 0, 55}] (* Robert G. Wilson v, Jun 09 2005 *)
s={};Do[id=IntegerDigits[n];ev=Select[id, EvenQ];ne=Select[id, OddQ];fd=FromDigits[{Length[ev], Length[ne], Length[id]}]; s=Append[s, fd], {n, 81}];SameQ[newA073053-s] (* Zak Seidov *)
deneat[n_]:=Module[{idn=IntegerDigits[n]},FromDigits[Flatten[ IntegerDigits/@ {Count[ idn,?EvenQ],Count[ idn,?OddQ],Length[ idn]}]]] Array[ deneat,60,0]// Flatten (* Harvey P. Dale, Aug 13 2021 *)

def a(n):
    s = str(n)
    e = sum(1 for c in s if c in "02468")
    return int(str(e) + str(len(s)-e) + str(len(s)))
print([a(n) for n in range(54)]) # Michael S. Branicky, Jan 21 2022

Edited and corrected by Jason Earls and Robert G. Wilson v, Jun 03 2005

a(0) added by N. J. A. Sloane, May 12 2019

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

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

A368866 The smallest positive number such that 2^a(n) when written in base n contains adjacent equal digits.

Original entry on oeis.org

2, 2, 4, 5, 6, 3, 6, 12, 16, 14, 11, 15, 8, 4, 8, 23, 16, 14, 16, 21, 9, 17, 20, 14, 30, 27, 16, 15, 10, 5, 10, 29, 48, 14, 46, 19, 18, 15, 32, 36, 27, 36, 18, 12, 56, 41, 37, 24, 58, 22, 26, 46, 58, 40, 29, 24, 36, 14, 20, 18, 12, 6, 12, 60, 62, 50, 49, 50, 20, 35, 36, 55, 61, 52, 53, 77

Offset: 2

Scott R. Shannon, Jan 08 2024

In the first 10000 terms the largest value is a(9031) = 1924, with a corresponding power of 2 of approximately 1.52*10^579.

			a(2) = 2 as 2^2 = 4 written in base 2 = 100_2 which contains adjacent 0's.
a(6) = 6 as 2^6 = 64 written in base 6 = 144_6 which contains adjacent 4's.
a(10) = 16 as 2^16 = 65536 written in base 10 = 65536_10 which contains adjacent 5's.

Scott R. Shannon, Table of n, a(n) for n = 2..10000

Cf. A000079, A171901, A011557, A004642, A004643, A000866.

f:= proc(n) local k,L;
  for k from 1 do
    L:= convert(2^k,base,n);
    if member(0, L[2..-1]-L[1..-2]) then return k fi
  od
end proc:
map(f, [$2..100]); # Robert Israel, Jan 09 2024

from itertools import count
from sympy.ntheory.factor_ import digits
def A368866(n):
    k = 1
    for m in count(1):
        k <<= 1
        s = digits(k,n)[1:]
        if any(s[i]==s[i+1] for i in range(len(s)-1)):
            return m # Chai Wah Wu, Jan 08 2024

A364049 a(n) is the least k such that the base-n digits of 2^k are not all distinct.

Original entry on oeis.org

2, 2, 4, 5, 6, 3, 6, 11, 16, 14, 11, 12, 8, 4, 8, 15, 16, 12, 16, 18, 9, 17, 15, 14, 24, 13, 16, 15, 10, 5, 10, 19, 24, 14, 21, 15, 18, 15, 19, 17, 17, 28, 18, 12, 24, 23, 31, 24, 31, 20, 26, 44, 35, 33, 25, 18, 36, 14, 14, 18, 12, 6, 12, 23, 45, 37, 38, 24, 20, 35, 36, 26, 51, 31, 33, 47, 34, 34

Offset: 2

Robert Israel, Jul 03 2023

			a(10) = 16 because 2^16 = 65536 does not have all distinct digits in base 10, while 2^k does have all distinct digits for 1 <= k <= 15.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A004642, A004643, A000866, A004645, A004646, A004647, A001357, A000079, A011557, A364052, A364089.

f:= proc(n) local k,L;
  for k from 2 do
    L:= convert(2^k,base,n);
    if nops(L) <> nops(convert(L,set)) then return k fi
  od;
end proc:
map(f, [$2..100]);

from itertools import count
from sympy.ntheory import digits
def a(n): return next(k for k in count(2) if len(set(d:=digits(1<Michael S. Branicky, Jul 05 2023

A305395 Records in A073053.

Original entry on oeis.org

11, 101, 112, 202, 213, 303, 314, 404, 415, 505

Offset: 1

N. J. A. Sloane, Jun 25 2018

The record-holders are the powers of 2 written in base 4, A004643.

M. E. Coppenbarger, Iterations of a modified Sisyphus function, Fib. Q., 56 (No. 2, 2018), 130-141.

Cf. A073053, A004643.

cp's OEIS Frontend

A004685 Fibonacci numbers written in base 2.

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A004642 Powers of 2 written in base 3.

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A004656 Powers of 3 written in base 2.

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A073053 Apply DENEAT operator (or the Sisyphus function) to 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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A004668 Powers of 3 written in base 26. (Next term contains a non-decimal digit.)

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