A157970 -id:A157970 - cp's OEIS frontend

A157971 Odious twin locations: first members of pairs of consecutive odious numbers.

1, 7, 13, 21, 25, 31, 37, 41, 49, 55, 61, 69, 73, 81, 87, 93, 97, 103, 109, 117, 121, 127, 133, 137, 145, 151, 157, 161, 167, 173, 181, 185, 193, 199, 205, 213, 217, 223, 229, 233, 241, 247, 253, 261, 265, 273, 279, 285, 289, 295

Offset: 1

John W. Layman, Mar 10 2009

An odious number (A000069) is a nonnegative integer with an odd number of ones in its binary expansion.

In the reference it is shown that these odious twins alternate with the evil twins (see A157970), which are pairs of consecutive evil numbers (A001969) having even numbers of ones in their binary expansions.

			The sequence of odious numbers (A000069) begins 1,2,4,7,8,11,13,14,16,19,21,..., so the first few odious twins are at 1,7,13, ... .

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Chris Bernhardt, Evil twins alternate with odious twins, Math. Mag. 82 (2009), pp. 57-62; also on JSTOR.
Eric Weisstein's World of Mathematics, Odious Number.

Cf. A000069, A091855, A157970, A248057.

SequencePosition[Table[If[OddQ[DigitCount[n,2,1]],1,0],{n,300}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 05 2016 *)

lista(nn) = select(n->((hammingweight(n) % 2) && (hammingweight(n+1) % 2)), vector(nn, i, i)); \\ Michel Marcus, Jul 10 2014

a(n) = A248057(n) - 1. - Amiram Eldar, Jun 16 2025

Comment corrected by Jeff Aronson. - N. J. A. Sloane, Oct 04 2020

A198680 Multiples of 3 whose sum of base-3 digits are also multiples of 3.

Original entry on oeis.org

0, 15, 21, 33, 39, 45, 57, 63, 78, 87, 93, 99, 111, 117, 132, 135, 150, 156, 165, 171, 186, 189, 204, 210, 222, 228, 234, 249, 255, 261, 273, 279, 294, 297, 312, 318, 327, 333, 348, 351, 366, 372, 384, 390, 396, 405, 420, 426, 438, 444, 450, 462, 468, 483, 489, 495

Offset: 1

John W. Layman, Oct 28 2011

It appears that Sum[k^j, 0<=k<=2^n-1, k in A198680] = Sum[k^j, 0<=k<=2^n-1, k in A198681] = Sum[k^j, 0<=k<=2^n-1, k in A180682], for 0<=j<=n-1, which has been verified numerically in a number of cases. This is a generalization of Prouhet's Theorem (see the reference). To illustrate for j=3, we have Sum[k^3, 0<=k<=2^n-1, k in A198680] = {0, 0, 12636, 1108809, 94478400, 7780827681, 633724260624, 51425722195929, 4168024588857600,...}, Sum[k^3, 0<=k<=2^n-1, k in A198681] = {0, 27, 14580, 1095687, 94478400, 7780827681, 633724260624, 51425722195929, 4168024588857600,..., Sum[k^3, 0<=k<=2^n-1, k in A198682] = {0, 216, 7776, 1121931, 94478400, 7780827681, 633724260624, 51425722195929, 4168024588857600,...}, and it is seen that all three sums agree for n>=4=j+1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Bernhardt, Evil twins alternate with odious twins, Math. Mag. 82 (2009), pp. 57-62.
Eric Weisstein's World of Mathematics, Prouhet-Tarry-Escott Problem

Cf. A000069, A001969, A157971, A157970, A198681, A198682.

Select[3*Range[0,200],Divisible[Total[IntegerDigits[#,3]],3]&] (* Harvey P. Dale, May 31 2014 *)

a(n) = 3*A079498(n). - Charles R Greathouse IV, Nov 02 2011

Offset corrected by Amiram Eldar, Jan 05 2020

A248056 Positions of 0,0 in the Thue-Morse sequence (A010060).

Original entry on oeis.org

6, 10, 18, 24, 30, 34, 40, 46, 54, 58, 66, 72, 78, 86, 90, 96, 102, 106, 114, 120, 126, 130, 136, 142, 150, 154, 160, 166, 170, 178, 184, 190, 198, 202, 210, 216, 222, 226, 232, 238, 246, 250, 258, 264, 270, 278, 282, 288, 294, 298, 306, 312, 318, 326, 330

Offset: 1

Clark Kimberling, Sep 30 2014

Every positive integer lies in exactly one of these four sequences: A248056, A091855, A091855, A248057.

			Thue-Morse sequence:  0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,..., so that a(1) = 6 and a(2) = 10.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A010060, A091855, A091785, A157970, A248057.

z = 400; u = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* A010060 *)
v = Rest[u]
t1 = Table[If[u[[n]] == 0 && v[[n]] == 0, 1, 0], {n, 1, z}];
t2 = Table[If[u[[n]] == 0 && v[[n]] == 1, 1, 0], {n, 1, z}];
t3 = Table[If[u[[n]] == 1 && v[[n]] == 0, 1, 0], {n, 1, z}];
t4 = Table[If[u[[n]] == 1 && v[[n]] == 1, 1, 0], {n, 1, z}];
Flatten[Position[t1, 1]]  (* A248056 *)
Flatten[Position[t2, 1]]  (* A091855 *)
Flatten[Position[t3, 1]]  (* A091785 *)
Flatten[Position[t4, 1]]  (* A248057 *)
SequencePosition[ThueMorse[Range[400]],{0,0}][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 02 2020 *)

a(n) = 2*A091785(n) for n >= 1.

a(n) = A157970(n) + 1. - Amiram Eldar, Jun 16 2025

A248104 Positions of 0,1,0 in the Thue-Morse sequence (A010060).

Original entry on oeis.org

4, 11, 16, 19, 28, 35, 44, 47, 52, 59, 64, 67, 76, 79, 84, 91, 100, 107, 112, 115, 124, 131, 140, 143, 148, 155, 164, 171, 176, 179, 188, 191, 196, 203, 208, 211, 220, 227, 236, 239, 244, 251, 256, 259, 268, 271, 276, 283, 292, 299, 304, 307, 316, 319, 324

Offset: 1

Clark Kimberling, Oct 01 2014

Every positive integer lies in exactly one of these six sequences:
A248056 (positions of 0,0,1)
A248104 (positions of 0,1,0)
A157970 (positions of 1,0,0)
A157971 (positions of 0,1,1)
A248105 (positions of 1,0,1)
A248057 (positions of 1,1,0)
The terms of the sequence are the positions of the mean of the positions of the three numbers 0, 1, 0. - Harvey P. Dale, Jan 26 2019

			Thue-Morse sequence:  0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,..., so that a(1) = 4 and a(2) = 11.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A010060, A248056, A157970, A157971, A248105, A248057.

z = 600; u = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 13]; v = Rest[u]; w = Rest[v]; t1 = Table[If[u[[n]] == 0 && v[[n]] == 0 && w[[n]] == 1, 1, 0], {n, 1, z}];
t2 = Table[If[u[[n]] == 0 && v[[n]] == 1 && w[[n]] == 0, 1, 0], {n, 1, z}];
t3 = Table[If[u[[n]] == 1 && v[[n]] == 0 && w[[n]] == 0, 1, 0], {n, 1, z}];
t4 = Table[If[u[[n]] == 0 && v[[n]] == 1 && w[[n]] == 1, 1, 0], {n, 1, z}];
t5 = Table[If[u[[n]] == 1 && v[[n]] == 0 && w[[n]] == 1, 1, 0], {n, 1, z}];
t6 = Table[If[u[[n]] == 1 && v[[n]] == 1 && w[[n]] == 0, 1, 0], {n, 1, z}];
Flatten[Position[t1, 1]]  (* A248056 *)
Flatten[Position[t2, 1]]  (* A248104 *)
Flatten[Position[t3, 1]]  (* A157970 *)
Flatten[Position[t4, 1]]  (* A157971 *)
Flatten[Position[t5, 1]]  (* A248105 *)
Flatten[Position[t6, 1]]  (* A248057 *)
Mean/@SequencePosition[ThueMorse[Range[400]],{0,1,0}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 26 2019 *)

A248105 Positions of 1,0,1 in the Thue-Morse sequence (A010060).

Original entry on oeis.org

3, 12, 15, 20, 27, 36, 43, 48, 51, 60, 63, 68, 75, 80, 83, 92, 99, 108, 111, 116, 123, 132, 139, 144, 147, 156, 163, 172, 175, 180, 187, 192, 195, 204, 207, 212, 219, 228, 235, 240, 243, 252, 255, 260, 267, 272, 275, 284, 291, 300, 303, 308, 315, 320, 323

Offset: 1

Clark Kimberling, Oct 01 2014

Every positive integer lies in exactly one of these six sequences:
A248056 (positions of 0,0,1)
A248104 (positions of 0,1,0)
A157970 (positions of 1,0,0)
A157971 (positions of 0,1,1)
A248105 (positions of 1,0,1)
A248057 (positions of 1,1,0)

			Thue-Morse sequence:  0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,..., so that a(1) = 3 and a(2) = 12.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A010060, A248056, A157970, A157971, A248104, A248057.

z = 600; u = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 13]; v = Rest[u]; w = Rest[v]; t1 = Table[If[u[[n]] == 0 && v[[n]] == 0 && w[[n]] == 1, 1, 0], {n, 1, z}];
t2 = Table[If[u[[n]] == 0 && v[[n]] == 1 && w[[n]] == 0, 1, 0], {n, 1, z}];
t3 = Table[If[u[[n]] == 1 && v[[n]] == 0 && w[[n]] == 0, 1, 0], {n, 1, z}];
t4 = Table[If[u[[n]] == 0 && v[[n]] == 1 && w[[n]] == 1, 1, 0], {n, 1, z}];
t5 = Table[If[u[[n]] == 1 && v[[n]] == 0 && w[[n]] == 1, 1, 0], {n, 1, z}];
t6 = Table[If[u[[n]] == 1 && v[[n]] == 1 && w[[n]] == 0, 1, 0], {n, 1, z}];
Flatten[Position[t1, 1]]  (* A248056 *)
Flatten[Position[t2, 1]]  (* A248104 *)
Flatten[Position[t3, 1]]  (* A157970 *)
Flatten[Position[t4, 1]]  (* A157971 *)
Flatten[Position[t5, 1]]  (* A248105 *)
Flatten[Position[t6, 1]]  (* A248057 *)

A331830 Numbers k such that k and k + 1 are both negabinary evil numbers.

Original entry on oeis.org

7, 13, 19, 27, 31, 39, 45, 51, 55, 61, 67, 75, 79, 87, 93, 99, 107, 111, 117, 123, 127, 135, 141, 147, 155, 159, 167, 173, 179, 183, 189, 195, 203, 207, 213, 219, 223, 231, 237, 243, 247, 253, 259, 267, 271, 279, 285, 291, 299, 303, 309, 315, 319, 327, 333, 339

Offset: 1

Amiram Eldar, Jan 28 2020

			7 is a term since both 7 and 7 + 1 = 8 are negabinary evil numbers (A268272): 7 has 4 digits of 1 in its negabinary representation, 11011, 8 has 2 digits of 1 in its negabinary representation, 11000, and both 4 and 2 are even.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A157970, A268272, A331831.

negaBinWt[n_] := negaBinWt[n] = If[n==0, 0, negaBinWt[Quotient[n-1, -2]] + Mod[n, 2]]; evilNegaBinQ[n_] := EvenQ[negaBinWt[n]]; c = 0; k = 1; s = {}; v = Table[-1, {2}]; While[c < 60, If[evilNegaBinQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 1]]]; k++]; s

A381848 Sequence obtained by replacing 3-term subwords of A010060 by 0,1,2,3,4,5 as described in Comments.

Original entry on oeis.org

2, 5, 4, 1, 3, 0, 2, 5, 3, 0, 1, 4, 2, 5, 4, 1, 3, 0, 1, 4, 2, 5, 3, 0, 2, 5, 4, 1, 3, 0, 2, 5, 3, 0, 1, 4, 2, 5, 3, 0, 2, 5, 4, 1, 3, 0, 1, 4, 2, 5, 4, 1, 3, 0, 2, 5, 3, 0, 1, 4, 2, 5, 4, 1, 3, 0, 1, 4, 2, 5, 3, 0, 2, 5, 4, 1, 3, 0, 1, 4, 2, 5, 4, 1, 3, 0

Offset: 1

Clark Kimberling, May 28 2025

nonn

The six 3-term subwords of A010060 are 0,0,1; 0,1,0; 0,1,1; 1,0,0; 1,0,1; 1,1,0. These are coded as 0,1,2,3,4,5 respectively, and then these numbers replace the corresponding subwords in A010060. The positions of 0,1,2,3,4,5 are given by A248956, A248104, A157971, A157970, A248105, A248057, respectively.

			Starting with A010060 = (0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0,...), the successive 3-term subwords are 0,1,1; 1,1,0; 1,0,1; 0,1,0; 1,0,0 ..., which code as 2,5,4,1,3,... .

Cf. A010060, A383999, A248956, A248104, A157971, A157970, A248105, A248057.

Partition[ThueMorse[Range[0, 200]], 3, 1] /. Thread[{{0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {1, 0, 0}, {1, 0, 1}, {1, 1, 0}} -> {0, 1, 2, 3, 4, 5}]  (* Peter J. C. Moses, May 22 2025 *)

cp's OEIS Frontend

A157971 Odious twin locations: first members of pairs of consecutive odious numbers.

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A198680 Multiples of 3 whose sum of base-3 digits are also multiples of 3.

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A248056 Positions of 0,0 in the Thue-Morse sequence (A010060).

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A248104 Positions of 0,1,0 in the Thue-Morse sequence (A010060).

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A248105 Positions of 1,0,1 in the Thue-Morse sequence (A010060).

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A331830 Numbers k such that k and k + 1 are both negabinary evil numbers.

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A381848 Sequence obtained by replacing 3-term subwords of A010060 by 0,1,2,3,4,5 as described in Comments.

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