A094148 -id:A094148 - cp's OEIS frontend

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

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

A154155 Decimal expansion of log_4 (10).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.660964047443681173935159714744695087932415696512290306027378...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_4(m): A094148 (m=3), A153201 (m=5), A153460 (m=6), A153615 (m=7), this sequence, A154176 (m=11), A154197 (m=12), A154224 (m=13), A154464 (m=14), A154543 (m=15), A154849 (m=17), A154909 (m=18), A155004 (m=19), A155183 (m=20), A155545 (m=21), A155695 (m=22), A155818 (m=23), A155936 (m=24).

evalf(log[4](10),100); # Wesley Ivan Hurt, Jan 29 2017

RealDigits[Log[4,10],10,120][[1]] (* Harvey P. Dale, May 07 2013 *)

A153201 Decimal expansion of log_4 (5).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.1609640474436811739351597147446950879324156965122903060273...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
András Sebo, Le charme discret des mathématiques, Images des Mathématiques, CNRS, 2006 (look for Theorem 2 Lovász, 1978).
Index entries for transcendental numbers

Cf. decimal expansion of log_4(m): A094148 (m=3), this sequence, A153460 (m=6), A153615 (m=7), A154155 (m=10), A154176 (m=11), A154197 (m=12), A154224 (m=13), A154464 (m=14), A154543 (m=15), A154849 (m=17), A154909 (m=18), A155004 (m=19), A155183 (m=20), A155545 (m=21), A155695 (m=22), A155818 (m=23), A155936 (m=24).

RealDigits[Log[4, 5], 10, 100][[1]] (* Vincenzo Librandi, Aug 30 2013 *)

A153460 Decimal expansion of log_4 (6).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.2924812503605780907268694719739082543799072038462405302278...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_4(m): A094148 (m=3), A153201 (m=5), this sequence, A153615 (m=7), A154155 (m=10), A154176 (m=11), A154197 (m=12), A154224 (m=13), A154464 (m=14), A154543 (m=15), A154849 (m=17), A154909 (m=18), A155004 (m=19), A155183 (m=20), A155545 (m=21), A155695 (m=22), A155818 (m=23), A155936 (m=24).

RealDigits[Log[4, 6], 10, 100][[1]] (* Vincenzo Librandi, Aug 30 2013 *)

A153615 Decimal expansion of log_4 (7).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.4036774610288020537209846586159154043205133129830703918386...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_4(m): A094148 (m=3), A153201 (m=5), A153460 (m=6), this sequence, A154155 (m=10), A154176 (m=11), A154197 (m=12), A154224 (m=13), A154464 (m=14), A154543 (m=15), A154849 (m=17), A154909 (m=18), A155004 (m=19), A155183 (m=20), A155545 (m=21), A155695 (m=22), A155818 (m=23), A155936 (m=24).

RealDigits[Log[4,7],10,120][[1]] (* Harvey P. Dale, Feb 18 2013 *)

A154176 Decimal expansion of log_4 (11).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.7297158093186486280996815233628964793516157628408840356564...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_4(m): A094148 (m=3), A153201 (m=5), A153460 (m=6), A153615 (m=7), A154155 (m=10), this sequence, A154197 (m=12), A154224 (m=13), A154464 (m=14), A154543 (m=15), A154849 (m=17), A154909 (m=18), A155004 (m=19), A155183 (m=20), A155545 (m=21), A155695 (m=22), A155818 (m=23), A155936 (m=24).

RealDigits[Log[4, 11], 10, 100][[1]] (* Vincenzo Librandi, Aug 30 2013 *)

A154197 Decimal expansion of log_4 (12).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.7924812503605780907268694719739082543799072038462405302278...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_4(m): A094148 (m=3), A153201 (m=5), A153460 (m=6), A153615 (m=7), A154155 (m=10), A154176 (m=11), this sequence, A154224 (m=13), A154464 (m=14), A154543 (m=15), A154849 (m=17), A154909 (m=18), A155004 (m=19), A155183 (m=20), A155545 (m=21), A155695 (m=22), A155818 (m=23), A155936 (m=24).

RealDigits[Log[4, 12], 10, 100][[1]] (* Vincenzo Librandi, Aug 30 2013 *)

A154224 Decimal expansion of log_4 (13).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.8502198590705460801984063271283473668142182008955186847692...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_4(m): A094148 (m=3), A153201 (m=5), A153460 (m=6), A153615 (m=7), A154155 (m=10), A154176 (m=11), A154197 (m=12), this sequence, A154464 (m=14), A154543 (m=15), A154849 (m=17), A154909 (m=18), A155004 (m=19), A155183 (m=20), A155545 (m=21), A155695 (m=22), A155818 (m=23), A155936 (m=24).

RealDigits[Log[4, 13], 10, 100][[1]] (* Vincenzo Librandi, Aug 30 2013 *)

A154464 Decimal expansion of log_4 (14).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.9036774610288020537209846586159154043205133129830703918386...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_4(m): A094148 (m=3), A153201 (m=5), A153460 (m=6), A153615 (m=7), A154155 (m=10), A154176 (m=11), A154197 (m=12), A154224 (m=13), this sequence, A154543 (m=15), A154849 (m=17), A154909 (m=18), A155004 (m=19), A155183 (m=20), A155545 (m=21), A155695 (m=22), A155818 (m=23), A155936 (m=24).

RealDigits[Log[4, 14], 10, 100][[1]] (* Vincenzo Librandi, Aug 30 2013 *)

Equals 1/2 + A153615. - R. J. Mathar, Feb 01 2023

A154543 Decimal expansion of log_4 (15).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.9534452978042592646620291867186033423123229003585308362552...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_4(m): A094148 (m=3), A153201 (m=5), A153460 (m=6), A153615 (m=7), A154155 (m=10), A154176 (m=11), A154197 (m=12), A154224 (m=13), A154464 (m=14), this sequence, A154849 (m=17), A154909 (m=18), A155004 (m=19), A155183 (m=20), A155545 (m=21), A155695 (m=22), A155818 (m=23), A155936 (m=24).

RealDigits[Log[4,15],10,120][[1]] (* Harvey P. Dale, Jul 19 2011 *)

cp's OEIS Frontend

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

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A154155 Decimal expansion of log_4 (10).

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A153201 Decimal expansion of log_4 (5).

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A153460 Decimal expansion of log_4 (6).

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A153615 Decimal expansion of log_4 (7).

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A154176 Decimal expansion of log_4 (11).

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A154197 Decimal expansion of log_4 (12).

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A154224 Decimal expansion of log_4 (13).

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