A000069 -id:A000069 - cp's OEIS frontend

A248172 Partition of the positive integers on minimal blocks such that concatenation of numbers in each block is an odious number (A000069). Sequence lists the odious numbers obtained in this way.

1, 2, 345, 67, 8, 9101112, 13, 14, 1516, 1718, 19, 202122232425, 26, 2728, 2930, 31, 32, 3334, 35, 3637, 38, 3940, 41, 42, 4344454647, 4849, 50, 5152, 5354, 55, 56, 5758, 59, 6061, 62, 6364, 6566, 67, 686970, 7172737475, 76, 7778, 79, 808182, 8384, 8586, 87

Offset: 1

Vladimir Shevelev, Oct 03 2014

The numbers of the consecutive positive integers over blocks of the partition are 1,1,3,2,1,4,1,1,2,2,1,6,1,2,2,1,1,2,1,...

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000069 (odious), A001969 (evil), A248009, A248138, A248140, A248171 (similar, with evil).

More terms from Peter J. C. Moses, Oct 04 2014

A248478 Evil numbers (A001969) becoming odious (A000069) if any digit is deleted (zeros allowed).

Original entry on oeis.org

12, 17, 18, 24, 27, 48, 71, 72, 77, 78, 111, 113, 114, 116, 119, 141, 144, 149, 169, 216, 221, 222, 225, 226, 228, 252, 255, 281, 282, 288, 311, 325, 387, 411, 414, 441, 442, 444, 447, 449, 474, 479, 497, 525, 526, 550, 556, 559, 562, 619, 621, 622, 649, 674

Offset: 1

Vladimir Shevelev, Oct 07 2014

			149 is in the sequence, since 149 = 2^7 + 2^4 + 2^2 + 1 is evil, while 49, 19 and 14 are odious.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000069, A001969, A248642, A248644, A248659.

More terms from Peter J. C. Moses, Oct 11 2014

A248642 Odious numbers (A000069) remaining odious if any digit is deleted (zeros allowed).

Original entry on oeis.org

11, 14, 21, 22, 28, 41, 42, 44, 47, 74, 81, 82, 84, 87, 88, 131, 161, 164, 191, 193, 194, 211, 256, 261, 262, 321, 322, 328, 352, 355, 381, 382, 388, 419, 421, 422, 491, 494, 502, 522, 552, 555, 569, 611, 614, 641, 642, 644, 647, 704, 744, 749, 764, 769, 793

Offset: 1

Vladimir Shevelev, Oct 10 2014

An analog of the idea Wilson-Dale: A034302, A051362.

			161 is in the sequence, since the numbers 161,61,11,16 are odious.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000069, A248478, A248644, A248659.

odiousQ:=OddQ[First[DigitCount[#,2]]]&;
Select[Range[1000],odiousQ[#]&&Apply[And,Map[odiousQ[FromDigits[#]]&,Subsets[#,{Length[#]-1}]&[IntegerDigits[#]]]]&] (* Peter J. C. Moses, Oct 10 2014 *)

More terms from Peter J. C. Moses, Oct 10 2014

A248659 Odious numbers (A000069) becoming evil (A001969) if any digit is deleted (zeros allowed).

Original entry on oeis.org

35, 50, 55, 56, 59, 69, 93, 100, 127, 157, 158, 185, 200, 203, 230, 233, 234, 239, 290, 299, 309, 333, 334, 336, 339, 346, 400, 403, 405, 406, 433, 436, 453, 458, 460, 463, 465, 466, 468, 517, 518, 548, 577, 578, 583, 653, 665, 666, 668, 727, 757, 758, 772

Offset: 1

Vladimir Shevelev, Oct 11 2014

			127 is in the sequence, since it is odious, while numbers 27,17,12 are evil.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000069, A001969, A248478, A248642, A248644.

odiousQ:=OddQ[First[DigitCount[#,2]]]&;
Select[Range[10,1000],odiousQ[#]&&Apply[And,Map[!odiousQ[FromDigits[#]]&,Subsets[#,{Length[#]-1}]&[IntegerDigits[#]]]]&] (* Peter J. C. Moses, Oct 11 2014 *)

More terms from Peter J. C. Moses, Oct 11 2014

A276444 Permutation of natural numbers: a(1) = 1; a(A001969(1+n)) = A088359(a(n)), a(A000069(1+n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 14, 15, 13, 16, 17, 18, 21, 19, 24, 27, 22, 20, 26, 30, 25, 31, 28, 23, 29, 32, 33, 34, 38, 35, 42, 48, 39, 36, 45, 54, 43, 58, 49, 40, 51, 37, 47, 57, 46, 62, 55, 44, 56, 63, 59, 50, 60, 41, 53, 61, 52, 64, 65, 66, 71, 67, 76, 86, 72, 68, 80, 96, 77, 106, 87, 73, 90, 69, 83, 102, 81

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276443.
Cf. A000069, A001969, A004001, A010060, A087686, A088359, A115384, A245710.
Similar or related permutations: A006068, A276442, A276446.

(definec (A276444 n) (cond ((< n 2) n) ((zero? (A010060 n)) (A088359 (A276444 (A245710 n)))) (else (A087686 (+ 1 (A276444 (- (A115384 n) 1)))))))

a(1) = 1, and for n > 1, if A010060(n) = 0 [when n is one of the evil numbers, A001969], a(n) = A088359(a(A245710(n))), otherwise a(n) = A087686(1+a(A115384(n)-1)).
As a composition of other permutations:
a(n) = A276442(A006068(n)).

A276446 Permutation of natural numbers: a(1) = 1; a(A000069(1+n)) = A088359(a(n)), a(A001969(1+n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 5, 11, 14, 15, 13, 8, 9, 10, 12, 20, 26, 30, 25, 31, 28, 23, 29, 16, 17, 18, 21, 19, 24, 27, 22, 37, 47, 57, 46, 62, 55, 44, 56, 63, 59, 50, 60, 41, 53, 61, 52, 32, 33, 34, 38, 35, 42, 48, 39, 36, 45, 54, 43, 58, 49, 40, 51, 70, 85, 105, 84, 120, 103, 82, 104, 126, 117, 98, 118, 79, 101, 119, 100, 127, 122, 108, 123, 89

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276445.
Cf. A000069, A001969, A004001, A010060, A087686, A088359, A115384, A245710.
Similar or related permutations: A006068, A267112, A276444.

(definec (A276446 n) (cond ((< n 2) n) ((zero? (A010060 n)) (A087686 (+ 1 (A276446 (A245710 n))))) (else (A088359 (A276446 (- (A115384 n) 1))))))

a(1) = 1, and for n > 1, if A010060(n) = 1 [when n is one of the odious numbers, A000069], a(n) = A088359(a(A115384(n)-1)), otherwise a(n) = A087686(1+a(A245710(n))).
As a composition of other permutations:
a(n) = A267112(A006068(n)).

A277902 If A010060(n) = 1, a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).

Original entry on oeis.org

1, 2, 3, 7, 6, 5, 4, 14, 9, 10, 13, 15, 8, 11, 12, 31, 24, 23, 28, 30, 25, 26, 17, 29, 16, 19, 18, 22, 27, 20, 21, 62, 43, 40, 61, 45, 56, 59, 54, 58, 49, 50, 33, 55, 36, 39, 52, 63, 32, 35, 48, 38, 57, 46, 37, 47, 34, 53, 44, 60, 41, 42, 51, 127, 102, 85, 124, 120, 121, 122, 83, 95, 112, 115, 68, 118, 89, 106

Offset: 1

Antti Karttunen, Nov 03 2016

a(n) gives the number that is in the same position in array A277880 as where n is located in array A277820.

			The top left corner of array A277820 is:
   1,  3,  5, 15
   2,  6, 10, 30
   7,  9, 27, 45
   4, 12, 20, 60
  13, 23, 57, 75
while the top left corner of A277880 is:
   1,  3,  6, 12
   2,  5, 10, 20
   4,  9, 18, 36
   7, 15, 30, 60
   8, 17, 34, 68
thus for example, a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 7, a(5) = 6, a(6) = 5, a(7) = 4, a(9) = 9, a(12) = 15, a(13) = 8 and a(27) = 18.

Inverse: A277901.
Cf. A000069, A001969, A001317, A003945, A006068, A010060, A065621, A129771, A268671, A277823, A277825.
Related permutations and arrays: A277820, A277821, A277880.

If A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).
As a composition of other permutations:
a(n) = A277880(A277821(n)).
Other identities. For all n >= 1:
A010060(a(n)) = A010060(n). [Preserves the parity of binary weight.]
a(A001317(n)) = A003945(n).
a(A065621(n)) = A000069(n).
a(A277823(n)) = A129771(n).
a(A277825(n)) = 2*A129771(n).

A290090 a(n) is the number of proper divisors of n that are odious (A000069).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 03 2017

If n is odd and k >= 1, then a(2^k*n) = (k+1)*n+k if n is in A000069 and (k+1)*n if n is not in A000069. - Robert Israel, Oct 03 2017

			For n = 55 whose proper divisors are 1, 5 and 11 (in binary "1", "101" and "1011"), only 1 and 11 have an odd number of 1's in their binary representations, thus a(55) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16385
Index entries for sequences related to binary expansion of n

Cf. A000005, A000069, A010060, A227872, A292257, A293231.

f:= proc(n) nops(select(t -> convert(convert(t,base,2),`+`)::odd, numtheory:-divisors(n) minus {n})) end proc:
map(f, [$1..200]); # Robert Israel, Oct 03 2017

Table[DivisorSum[n, 1 &, And[OddQ@ DigitCount[#, 2, 1], # < n] &], {n, 120}] (* Michael De Vlieger, Oct 03 2017 *)

PARI
```
A290090(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA010060(d).
a(n) = A227872(n) - A010060(n).
a(n) = A007814(A293231(n)).
A000035(a(n)) = A000035(A292257(n)). [Parity-wise equivalent with A292257.]

A341458 Unique square array T(n, k) read by antidiagonals, n, k > 0, such that A000069(T(n, k)) = A341288(A000069(n), A000069(k)).

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 5, 5, 4, 5, 6, 1, 6, 5, 6, 3, 7, 7, 3, 6, 7, 4, 2, 1, 2, 4, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 7, 4, 2, 1, 2, 4, 7, 9, 10, 17, 6, 3, 7, 7, 3, 6, 17, 10, 11, 18, 33, 5, 6, 1, 6, 5, 33, 18, 11, 12, 22, 39, 57, 4, 5, 5, 4, 57, 39, 22, 12

Offset: 1

Rémy Sigrist, Feb 12 2021

The positive integers equipped with T form a group.

Every row (and column) is a self-inverse permutation of the positive integers.

			Array T(n, k) begins:
  n\k|   1   2   3   4   5   6    7    8    9   10   11   12   13   14   15   16
  ---+--------------------------------------------------------------------------
    1|   1   2   3   4   5   6    7    8    9   10   11   12   13   14   15   16
    2|   2   1   5   6   3   4    8    7   17   18   22   21   20   19   23   24
    3|   3   5   1   7   2   8    4    6   33   39   35   37   36   38   34   40
    4|   4   6   7   1   8   2    3    5   57   63   62   60   61   59   58   64
    5|   5   3   2   8   1   7    6    4   65   71   70   68   69   67   66   72
    6|   6   4   8   2   7   1    5    3   89   95   91   93   92   94   90   96
    7|   7   8   4   3   6   5    1    2  105  106  110  109  108  107  111  112
    8|   8   7   6   5   4   3    2    1  113  114  115  116  117  118  119  120
    9|   9  17  33  57  65  89  105  113    1   25   41   49   73   81   97  121
   10|  10  18  39  63  71  95  106  114   25    1   56   48   88   80  121   97
   11|  11  22  35  62  70  91  110  115   41   56    1   32  104  121   80   81
   12|  12  21  37  60  68  93  109  116   49   48   32    1  121  104   88   73
   13|  13  20  36  61  69  92  108  117   73   88  104  121    1   32   48   49
   14|  14  19  38  59  67  94  107  118   81   80  121  104   32    1   56   41
   15|  15  23  34  58  66  90  111  119   97  121   80   88   48   56    1   25
   16|  16  24  40  64  72  96  112  120  121   97   81   73   49   41   25    1

Rémy Sigrist, Table of n, a(n) for n = 1..8385
Rémy Sigrist, Colored representation of the table for 1 <= n, k <= 128
Rémy Sigrist, PARI program for A341458

Cf. A000069, A341288.

See A341487 and A341489 for the second and third rows, respectively.

PARI
```
See Links section.
```

T(n, k) = T(k, n).
T(n, 1) = n.
T(n, n) = 1.

A380008 Numbers t whose binary expansion Sum 2^e_i has exponents e_i which are odious numbers (A000069).

Original entry on oeis.org

0, 2, 4, 6, 16, 18, 20, 22, 128, 130, 132, 134, 144, 146, 148, 150, 256, 258, 260, 262, 272, 274, 276, 278, 384, 386, 388, 390, 400, 402, 404, 406, 2048, 2050, 2052, 2054, 2064, 2066, 2068, 2070, 2176, 2178, 2180, 2182, 2192, 2194, 2196, 2198, 2304, 2306, 2308, 2310, 2320, 2322, 2324, 2326, 2432, 2434, 2436, 2438, 2448, 2450, 2452, 2454

Offset: 0

Luis Rato, Jan 08 2025

These t in binary representation have 1s only in positions with 0s in the Thue-Morse sequence (A010059) with beginning of that sequence corresponding to least significant bit. a(n) can be derived from n by placing the bits of n into a(n) at those permitted positions.
a(n) can be represented in base 4 equal to binary representation of n with each digit multiplied by 1 or 2 according to the 1-2 Thue-Morse sequence A001285 starting in the least significant digit and transforming 1->2, and 2->1.
Any pair 2*p and 2*p+1 has one evil and the other odious number, so the bit at position p in n goes to either 2*p or 2*p+1 in a(n), according as which of those is odious.
Every integer k>=0 corresponds to a unique pair i,j with k = x(i) + y(j), with x(i)=a(i) and y(j)=A380009(j).
Sequences x(n) and y(n) have same growth rate and cross an infinite number of times.
Coordinate pairs (i,j), define a Morton space-filling curve, similar to Z-order curve.

			Considering the representation in base 4,
For n=11 = 1011_binary, a(11) -> 1021_base4 -> 2012_base4 = 134.
For n=12 = 1100_binary, a(12) -> 1200_base4 -> 2100_base4 = 144.
Considering all numbers are decomposed in binary, with exponents belonging to odious numbers: 1, 2, 4, 7,...
The sequence of terms together with their binary representation begins:
 n    a(n)      a(n)_bin
 0     0:         0 ~               0
 1     2:        10 ~             2^1
 2     4:       100 ~         2^2
 3     6:       110 ~         2^2+2^1
 4    16:     10000 ~     2^4
 5    18:     10010 ~     2^4   +2^1
 6    20:     10100 ~     2^4+2^2
 7    22:     10110 ~     2^4+2^2+2^1
 8   128:  10000000 ~ 2^7
 9   130:  10000010 ~ 2^7        +2^1
10   132:  10000100 ~ 2^7    +2^2
11   134:  10000110 ~ 2^7    +2^2+2^1
12   144:  10010000 ~ 2^7+2^4

Luis Rato, Plot of an NZ-order curve, containing the integers from 0 to 255.
Wikipedia, Morton code map, also known as Z-order curve.

Cf. A000069, A001285, A010059, A380009.

a(n) = { my (v = 0, e); while (n, n -= 2^e = exponent(n); v += 2^(2*e + if (hammingweight(e)%2, 0, 1));); return (v); } \\ Rémy Sigrist, Feb 02 2025

isok(t) = my(b=Vecrev(binary(t))); for (i=1, #b, if (b[i] && !(hammingweight(i-1)%2), return(0))); return(1); \\ Michel Marcus, Feb 10 2025

cp's OEIS Frontend

A248172 Partition of the positive integers on minimal blocks such that concatenation of numbers in each block is an odious number (A000069). Sequence lists the odious numbers obtained in this way.

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A248478 Evil numbers (A001969) becoming odious (A000069) if any digit is deleted (zeros allowed).

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A248642 Odious numbers (A000069) remaining odious if any digit is deleted (zeros allowed).

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A248659 Odious numbers (A000069) becoming evil (A001969) if any digit is deleted (zeros allowed).

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A276444 Permutation of natural numbers: a(1) = 1; a(A001969(1+n)) = A088359(a(n)), a(A000069(1+n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

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A276446 Permutation of natural numbers: a(1) = 1; a(A000069(1+n)) = A088359(a(n)), a(A001969(1+n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

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A277902 If A010060(n) = 1, a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).

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A290090 a(n) is the number of proper divisors of n that are odious (A000069).

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A341458 Unique square array T(n, k) read by antidiagonals, n, k > 0, such that A000069(T(n, k)) = A341288(A000069(n), A000069(k)).

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