A004656 -id:A004656 - cp's OEIS frontend

A004663 Powers of 3 written in base 9.

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

A037097 Periodic vertical binary vectors of powers of 3, starting from bit-column 2 (halved).

Original entry on oeis.org

0, 12, 120, 57120, 93321840, 10431955353116229600, 8557304989566294213168677685339060480, 102743047168201563425402150421568484707810385382513037790885688657488312400960

Offset: 2

Antti Karttunen, Jan 29 1999

Conjecture: For n>=3, each term a(n), when considered as a GF(2)[X]-polynomial, is divisible by GF(2)[X] -polynomial (x + 1) ^ A000225(n-1) (= A051179(n-2)). If this holds, then for n>=3, a(n) = A048720bi(A136386(n),A048723bi(3,A000225(n-1))) = A048720bi(A136386(n),A051179(n-2)).

			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, starting from the column 2 from the right, the bits in the n-th column can be arranged in periods of 2^(n-1): 4, 8, ... This sequence is formed from those bits: 0011, reversed is 11100, which is binary for 12, thus a(3) = 12, 00011110, reversed is 011110000, which is binary for 120, thus a(4) = 120.

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

A. Karttunen, Table of n, a(n) for n = 2..12
A. Karttunen, C program for computing this sequence
S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

a(n) = floor(A037096(n)/(2^(2^(n-1)))). See also A036284, A136386.

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

a(n) = Sum_{k=0..A000225(n-1)} ([A000244(k)/(2^n)] mod 2) * 2^k, where [] stands for floor function.

Entry revised Dec 29 2007

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

A004669 Powers of 3 written in base 27.

Original entry on oeis.org

1, 3, 9, 10, 30, 90, 100, 300, 900, 1000, 3000, 9000, 10000, 30000, 90000, 100000, 300000, 900000, 1000000, 3000000, 9000000, 10000000, 30000000, 90000000, 100000000, 300000000, 900000000, 1000000000

Offset: 0

N. J. A. Sloane

Similar to powers of 2 in base 8 (A004647) or 16 (A004655). - M. F. Hasler, Jun 22 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,10).

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.

Table[FromDigits[IntegerDigits[3^n, 27]], {n, 0, 100}] (* G. C. Greubel, Oct 12 2018 *)

apply( a(n)=3^(n%3)*10^(n\3), [0..20]) \\ M. F. Hasler, Jun 22 2018

a(n) = 3^(n mod 3)*10^floor(n/3). - M. F. Hasler, Jun 22 2018
From Chai Wah Wu, Sep 03 2020: (Start)
a(n) = 10*a(n-3) for n > 2.
G.f.: (-9*x^2 - 3*x - 1)/(10*x^3 - 1). (End)

A211864 Powers of three read in base 2.

Original entry on oeis.org

1, 3, 9, 11, 17, 19, 41, 43, 81, 131, 169, 219, 241, 419, 681, 651, 849, 1203, 2441, 1867, 4369, 3027, 5625, 12475, 15457, 26403, 26553, 48795, 54561, 99667, 80345, 158731, 221313, 332467, 460857, 375451, 569345, 987267, 1181145, 1594971, 1924001, 4458531

Offset: 0

Robert G. Wilson v, Feb 10 2013

			a(3)=11 because 3^3=27_10 but in base 2 it is interpreted as 2*2^1+7*2^0=4+7=11. That is to say, digits larger than the base are carried.

Cf. A004656.

f[n_] := FromDigits[ IntegerDigits[3^n], 2]; Array[f, 42, 0]

A385157 Numbers k so that the binary expansion of 3^k starts with the binary expansion of k.

Original entry on oeis.org

1, 2, 3, 9, 27, 65, 95, 123, 163, 303, 451, 597, 760, 1757, 2546, 2700, 7142, 25030, 25719, 25772, 49105, 61426, 90981, 107497, 194210, 659077, 6732590, 8513462, 9344030, 14549893, 32276115, 89912342, 181720904, 280120681, 437484689, 896754175, 10625891495, 30605576222

Offset: 1

Jayde S. Massmann, Jun 19 2025

			9 is in the sequence as 3^9 is 100110011100011 in binary, and 9 is 1001.

Cf. A000244, A004656.

q[k_] := k < Log[3, k+1] + (Floor[k*Log2[3]-Log2[k]])/Log2[3]; Select[Range[10^5], q] (* Amiram Eldar, Jun 20 2025 *)

isok(k) = my(bk = binary(k), vb=Vec(binary(3^k), #bk)); vb == bk; \\ Michel Marcus, Jun 20 2025

a(26) from Hugo Pfoertner, Jun 20 2025
a(27)-a(36) from Amiram Eldar, Jun 20 2025
a(37)-a(38) from Jinyuan Wang, Jun 27 2025

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A004663 Powers of 3 written in base 9.

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A037097 Periodic vertical binary vectors of powers of 3, starting from bit-column 2 (halved).

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

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

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A004669 Powers of 3 written in base 27.

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A211864 Powers of three read in base 2.

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A385157 Numbers k so that the binary expansion of 3^k starts with the binary expansion of k.