A223024 -id:A223024 - cp's OEIS frontend

A224072 Odd odious numbers divisible by 3.

21, 69, 81, 87, 93, 117, 171, 213, 261, 273, 279, 285, 309, 321, 327, 333, 339, 345, 351, 357, 369, 375, 381, 405, 453, 465, 471, 477, 501, 555, 597, 651, 675, 681, 687, 699, 747, 789, 837, 849, 855, 861, 885, 939, 981, 1029, 1041, 1047, 1053, 1077, 1089, 1095

Offset: 1

Vladimir Shevelev, Mar 30 2013

By Moser-Newman phenomenon among the first N positive integers multiple of 3, the evil numbers are always in the majority. Moreover, this excess tends to infinity as N goes to infinity and its growth is of order N^a, where a = log(3)/log(4).

Peter J. C. Moses, Table of n, a(n) for n = 1..10000
J. Coquet, A summation formula related to the binary digits, Inventiones Mathematicae 73 (1983), pp. 107-115.
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.
Vladimir Shevelev, Generalized Newman phenomena and digit conjectures on primes, Internat. J. of Mathematics and Math. Sciences, 2008 (2008), Article ID 908045, 1-12.

Cf. A000069, A001969, A223024, A180963.

Select[Range[3, 2000, 6], OddQ[DigitCount[#, 2]][[1]] &] (* Peter J. C. Moses, Apr 04 2013 *)

isok(m) = (m % 2) && !(m % 3) && (hammingweight(m) % 2); \\ Michel Marcus, Feb 20 2021

A237545 Odious powers of 3.

Original entry on oeis.org

1, 81, 2187, 59049, 177147, 1594323, 14348907, 43046721, 1162261467, 3486784401, 31381059609, 22876792454961, 68630377364883, 205891132094649, 16677181699666569, 150094635296999121, 36472996377170786403, 328256967394537077627, 8862938119652501095929, 79766443076872509863361

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Feb 09 2014

Intersection of A000069 and A000244.
Exponents of a(n): A223024.
It seems that this sequence includes about half of the powers of 3. For example, a(50171) = 3^99999. - Charles R Greathouse IV, Mar 05 2014

Robert Israel, Table of n, a(n) for n = 1..1058

Cf. A000069 (odious numbers), A000244 (powers of 3), A223024.

select(t -> convert(convert(t,base,2),`+`)::odd, [seq(3^i,i=0..100)]); # Robert Israel, Oct 10 2016

Select[3^Range[32],OddQ[First[DigitCount[#,2] ] ]&] (* Wouter Meeussen, Feb 09 2014 *)

for(n=0,100,if(hammingweight(t=3^n)%2,print1(t", "))) \\ Charles R Greathouse IV, Mar 05 2014

Corrected and more terms added by Robert Israel, Oct 10 2016

A371970 Exponents k such that the binary expansion of 3^k has an even number of ones.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 12, 14, 17, 18, 21, 23, 24, 25, 26, 27, 31, 32, 33, 35, 37, 38, 39, 40, 42, 44, 45, 47, 51, 52, 55, 57, 58, 59, 60, 61, 64, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 96, 99, 102, 104, 105, 106, 109, 112, 116, 127, 131, 132, 133, 134, 135, 136

Offset: 1

Hugo Pfoertner, Apr 24 2024

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Complement of A223024.

Cf. A000120, A000244, A001969, A011754, A078839.

q:= n-> is(add(i, i=Bits[Split](3^n))::even):
select(q, [$0..150])[];  # Alois P. Heinz, Apr 24 2024

Select[Range[136], EvenQ@ DigitCount[3^#, 2, 1] &] (* Michael De Vlieger, Apr 24 2024 *)

is_a371970(k) = hammingweight(3^k)%2 == 0

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A224072 Odd odious numbers divisible by 3.

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A237545 Odious powers of 3.

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A371970 Exponents k such that the binary expansion of 3^k has an even number of ones.

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