A180963 -id:A180963 - cp's OEIS frontend

A036556 Integers which when multiplied by 3 have an odd number of 1's in their binary expansion (cf. A000069).

7, 14, 23, 27, 28, 29, 31, 39, 46, 54, 56, 57, 58, 62, 71, 78, 87, 91, 92, 93, 95, 103, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 119, 123, 124, 125, 127, 135, 142, 151, 155, 156, 157, 159, 167, 174, 182, 184, 185, 186, 190, 199, 206, 214

Offset: 1

Dan Hoey

In other words, numbers n such that 3n is odious.
Numbers n such that valuation(binomial(3n,n),2) is odd. - Benoit Cloitre, Jun 06 2004
Intersection of A000069 and A008585 (multiples of 3), divided by 3.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000069, A036555, A036557, A180963.

[ n : n in [0..150] | IsOdd(&+Intseq(3*n, 2))]; // Vincenzo Librandi, Apr 13 2011

Select[ Range[ 214 ], OddQ[ Plus@@IntegerDigits[ 3#, 2 ]]& ]

for(n=1,214,if(valuation(binomial(3*n,n),2)%2==1,print1(n,","))) \\ Benoit Cloitre, Jun 06 2004

a(n) = A180963(n)/3. - Amiram Eldar, Aug 06 2023

Definition corrected by N. J. A. Sloane, Jan 09 2007

A224072 Odd odious numbers divisible by 3.

Original entry on oeis.org

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

A360799 Numbers m with m mod 3 = q, q != 2, such that the number of ones in its base-2 representation is even if q=0 and odd if q=1.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 12, 13, 15, 16, 18, 19, 22, 24, 25, 27, 28, 30, 31, 33, 36, 37, 39, 45, 48, 49, 51, 52, 54, 55, 57, 60, 61, 63, 64, 66, 67, 70, 72, 73, 75, 76, 78, 79, 82, 88, 90, 91, 94, 96, 97, 99, 100, 102, 103, 105, 108, 109, 111, 112, 114, 115, 118, 120, 121

Offset: 0

Gerhard Kirchner, Feb 24 2023

nonn

For q=0, the terms in A180963 are excluded.
The terms of the sequence occur, with some exceptions, while tiling a wall (odd width w) with 1 X 2 dominos. The current tiling status can be described by a number x with 0 <= x < 2^w. In the base-2 representation, 1 stands for an overstanding unit square, see example.
Statement:
The tiling always starts with q=1 and an odd number of ones (type 1) and is followed by a term with q=0 and an even number of ones (type 2) and so on, alternately.
Proof:
Start, provisionally, with w upright dominos. The corresponding term is x = (11..1) = 2^w-1 with x mod 3 = 1 (type 1). Another first profile can be generated by replacing a pair of adjacent upright dominos with one flat domino. In the base-2 representation, this is the subtraction (11..11..1) - (00..11..0) = (11..00..1). The subtrahend is 3*2^j with 0 <= j < w. Therefore, the modified term also is type 1. This way, any first profile can be found and it is type 1.
In the next provisional step, an upright domino is placed on each not overstanding unit square. If p1 is the first profile, then the second is p2 = 2^w - 1 - p1 with p2 mod 3 = 0. Moreover, the transition from p1 to p2 exchanges the ones and zeros such that p2 is type 2. Again, replacing adjacent upright dominos by one flat domino does not change the type of the profile. The next profile is type 1 and so on. QED. Condition to be satisfied by a tiling profile: The continued removal of 00 and 11 (reduction) leads to (0) or (1). Example: a(10)=18=(10010) -> (110) -> (0). The first exceptions are a(314) = 682 = (01010101010), a(611) = 1365 = (10101010101) and a(988) = 2218 = (0100010101010). Note that the reduction of 2218 is 682.

			 5 X 4 wall is tiled bottom-up with 1 X 2 dominos:
                                      _    ___ ___ _
                 _ _          _ _ ___| |  |_ _|___| |
        _       | | |_ ___   | | |_ _|_|  | | |_ _|_|
    ___| |___   |_|_| |___|  |_|_| |___|  |_|_| |___|
   |___|_|___|  |___|_|___|  |___|_|___|  |___|_|___|
    0 0 1 0 0    1 1 0 0 0    0 0 0 0 1    0 0 0 0 0
     4 = a(3)   24 = a(14)     1 = a(1)     0 = a(0)

Cf. A001835, A003775, A028469, A028471, A180963, A187616, A360800.

block(kmax: 100, a:[],
 even_ones(x):= block(su:0,
  while x>0 do(p: mod(x,2), x:(x-p)/2, su:su+p),
   return(mod(su,2))),
for k from 0 thru kmax do(r:mod(k,3),
 if r<2 and r=even_ones(k) then a:append(a,[k])),a);

isok(m) = my(k=m%3); if (hammingweight(m) % 2, k==1, k==0); \\ Michel Marcus, Feb 27 2023

A360800 Numbers Sum_{i=1..2r+1} 2^k(i) such that k(1) is even and, for r > 0 and i < 2r+1, the difference k(i+1)-k(i) is > 0 and odd.

Original entry on oeis.org

1, 4, 7, 16, 19, 25, 28, 31, 64, 67, 73, 76, 79, 97, 100, 103, 112, 115, 121, 124, 127, 256, 259, 265, 268, 271, 289, 292, 295, 304, 307, 313, 316, 319, 385, 388, 391, 400, 403, 409, 412, 415, 448, 451, 457, 460, 463, 481, 484, 487, 496, 499, 505, 508, 511, 1024

Offset: 1

Gerhard Kirchner, Feb 24 2023

nonn

This is a subsequence of A360799. Another description of the terms: in the base-2 representation, the number of ones is odd and all zeros are grouped in blocks of even length.

That is why the terms less than 2^(2j+1) describe start profiles for tiling a (2j+1) X m wall with 1 X 2 dominos, see examples and A360799.

			A 5 X m wall is tiled bottom-up with dominos, start profiles:
            _        _            _ _ _    _     _ _    _ _ _ _ _
    ___ ___| |   ___| |___    ___| | | |  | |___| | |  | | | | | |
   |___|___|_|  |___|_|___|  |___|_|_|_|  |_|___|_|_|  |_|_|_|_|_|
    0 0 0 0 1    0 0 1 0 0    0 0 1 1 1    1 0 0 1 1    1 1 1 1 1
    1 = a(1)     4 = a(2)     7 = a(3)     19 = a(5)    31 = a(7)
    also the mirror images of 1 (16), 19 (25) and 7 (28).

Cf. A001835, A003775, A028469, A028471, A180963, A187616, A360799.

block(kmax: 100, a:[],
 oddsum(y):= block(su1:0, su2:0, pold:0, ok: true,
  while y>0 and ok do(p:mod(y,2), y:(y-p)/2,
   if p=1 then(if pold=0 and su2=1 then ok:false, su1:1-su1, su2:0)
   elseif p=0 then su2:1-su2, pold:p), return(is(ok and su1=1))),
for k from 1 thru kmax do if oddsum(k) then a:append(a,[k]),a);

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A036556 Integers which when multiplied by 3 have an odd number of 1's in their binary expansion (cf. A000069).

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A224072 Odd odious numbers divisible by 3.

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A360799 Numbers m with m mod 3 = q, q != 2, such that the number of ones in its base-2 representation is even if q=0 and odd if q=1.

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A360800 Numbers Sum_{i=1..2r+1} 2^k(i) such that k(1) is even and, for r > 0 and i < 2r+1, the difference k(i+1)-k(i) is > 0 and odd.

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