A004655 -id:A004655 - cp's OEIS frontend

A004642 Powers of 2 written in base 3.

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

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.

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

A004663 Powers of 3 written in base 9.

Original entry on oeis.org

1, 3, 10, 30, 100, 300, 1000, 3000, 10000, 30000, 100000, 300000, 1000000, 3000000, 10000000, 30000000, 100000000, 300000000, 1000000000, 3000000000, 10000000000, 30000000000, 100000000000, 300000000000, 1000000000000, 3000000000000, 10000000000000, 30000000000000

Offset: 0

N. J. A. Sloane

Robert Israel, Table of n, a(n) for n = 0..1997
Index entries for linear recurrences with constant coefficients, signature (0,10).

Cf. A026383, A016116.
Cf. A000244, A004656, A004658, A004659, ... : powers of 3 in base 10, 2, 4, 5, ...
Cf. A000079, A004642, ..., A004655: powers of 2 written in base 10, 2, 3, ..., 16.

seq(op([10^i,3*10^i]),i=0..100); # Robert Israel, Jun 25 2018

Table[FromDigits[IntegerDigits[3^n, 9]], {n, 0, 100}] (* G. C. Greubel, Oct 12 2018 *)

a(n)=3^bittest(n,0)*10^(n\2) \\ M. F. Hasler, Jun 25 2018

From Paul Barry, Jul 14 2004: (Start)
G.f.: (1 + 3*x)/(1 - 10*x^2);
a(n) = 2*a(n-1) + 3*a(n-2) + 10^floor((n-2)/2);
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2), k)*3^(n-2*k). (End)
a(n) = 3*a(n-1) + ((1 + (-1)^n)/2)*a(n-2) with a(0)=1, a(1)=3. - Taras Goy, Mar 20 2019
E.g.f.: cosh(sqrt(10)*x) + 3*sinh(sqrt(10)*x)/sqrt(10). - Stefano Spezia, Mar 31 2023

A004645 Powers of 2 written in base 6.

Original entry on oeis.org

1, 2, 4, 12, 24, 52, 144, 332, 1104, 2212, 4424, 13252, 30544, 101532, 203504, 411412, 1223224, 2450452, 5341344, 15123132, 34250304, 112541012, 225522024, 455444052, 1355332144, 3155104332, 10354213104

Offset: 0

N. J. A. Sloane

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

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, ...

[Seqint(Intseq(2^n, 6)): n in [0..30]]; // G. C. Greubel, Sep 10 2018

Table[FromDigits[IntegerDigits[2^n, 6]], {n, 0, 40}] (* Vincenzo Librandi, Jun 07 2013 *)

a(n)=fromdigits(digits(2^n,6)) \\ M. F. Hasler, Jun 23 2018

A004653 Powers of 2 written in base 14. (Next term contains a non-decimal character.)

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 48, 92, 144, 288, 532

Offset: 0

N. J. A. Sloane

Next term contains a non-decimal character if such characters are chosen to represent digits > 9, where "digit" means the coefficients in N = Sum_{k>=0} d_k * b^k. This isn't possible here, but digits 0, 10, ..., 13 could be represented, e.g., using 00, 10, ..., 40. This would not affect a(0)..a(10), which don't have a digit 0. - M. F. Hasler, Jun 25 2018

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

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

BaseForm[Table[2^n, {n, 0, 10}], 14] (* Alonso del Arte, Mar 18 2005 *)

apply( a(n)=fromdigits(digits(2^n,14)), [0..10]) \\ This yields Sum d[k]*10^k where d[k] are the base 14 digits. To get strings possibly containing letters 'A'..'D' replace fromdigits(...) by Strchr(apply(d->48+d+(d>9)*7,...)). - M. F. Hasler, Jun 25 2018

A004654 Powers of 2 written in base 15. (Next term contains a non-decimal character.)

Original entry on oeis.org

1, 2, 4, 8, 11, 22, 44, 88, 121, 242, 484, 918, 1331, 2662

Offset: 0

N. J. A. Sloane

a(14) is defined as "2^14 written in base 15". However, the base-15 digits of 2^14 are [4,12,12,4]. The standard convention of using letters A, B, ... to represent digits > 9, cannot be used in the Data sections of OEIS entries. One possibility to encode such terms within the given constraints and without affecting the earlier terms would be to use 00, 10, 20, ..., 50 to represent unambiguously the digits 0, 10, 11, ..., 14. - M. F. Hasler, Jun 22 2018.

This sequence makes a nice puzzle. It would be possible to allow non-decimal characters by using pairs of decimal digits instead of single digits, but this would spoil the beauty of the puzzle. - N. J. A. Sloane, Jun 25 2018

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

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

apply( a(n,b=15,m=2)=fromdigits(digits(m^n,b)), [0..13]) \\ This sums the base-15 digits multiplied by powers of 10. Digits > 9 occurring for n >= 14 will carry over to the left (4CC4 -> 5324). - M. F. Hasler, Jun 22 2018

A004667 Powers of 3 written in base 13. (Next term contains a non-decimal digit.)

Original entry on oeis.org

1, 3, 9, 21, 63, 159, 441

Offset: 0

N. J. A. Sloane

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

Cf. A000244, A004656, A004658, A004659, ..., A004668: powers of 3 in base 10, 2, 4, 5, ..., 26.

FromDigits[IntegerDigits[#,13]]&/@(3^Range[0,6]) (* Harvey P. Dale, Mar 05 2018 *)

apply( a(n, b=13, m=3)=fromdigits(digits(m^n, b)), [0..6]) \\ This implements one possible continuation of the sequence beyond n = 6: write digits in decimal and carry over (so CC4 -> 12*100 + 12*10 + 4 = 1324). - M. F. Hasler, Jun 22 2018

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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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Magma

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

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A004663 Powers of 3 written in base 9.

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A004645 Powers of 2 written in base 6.

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A004653 Powers of 2 written in base 14. (Next term contains a non-decimal character.)

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