

A011754


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


8



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
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OFFSET

0,2


COMMENTS

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


REFERENCES

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


LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000
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, 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 268280.
H. G. Senge and E. G. Straus, PVnumbers and sets of multiplicity, Periodica Mathematica Hungarica 3 (1973), pp. 93100.


FORMULA

a(n) = A000120(3^n).  Benoit Cloitre, Dec 06 2002
a(n) = A000120(A000244(n)).  Reinhard Zumkeller, Aug 14 2015


MATHEMATICA

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 *)


PROG

(Haskell)
a011754 = a000120 . a000244  Reinhard Zumkeller, Aug 14 2015
(PARI) a(n)=hammingweight(3^n) \\ Charles R Greathouse IV, Feb 09 2017
(Magma) [&+Intseq(3^n, 2): n in [0..79]]; // Vincenzo Librandi, Nov 28 2018


CROSSREFS

Cf. A007088, A000120, A000244, A004656, A261009, A001370, A094148.
Sequence in context: A275234 A301768 A088145 * A090105 A082146 A037145
Adjacent sequences: A011751 A011752 A011753 * A011755 A011756 A011757


KEYWORD

nonn,nice,easy


AUTHOR

Allan C. Wechsler, Dec 11 1999


EXTENSIONS

More terms from Stefan Steinerberger, Apr 03 2006


STATUS

approved



