A157971 -id:A157971 - cp's OEIS frontend

A158704 Nonnegative integers with an even number of even powers of 2 in their base-2 representation.

0, 2, 5, 7, 8, 10, 13, 15, 17, 19, 20, 22, 25, 27, 28, 30, 32, 34, 37, 39, 40, 42, 45, 47, 49, 51, 52, 54, 57, 59, 60, 62, 65, 67, 68, 70, 73, 75, 76, 78, 80, 82, 85, 87, 88, 90, 93, 95, 97, 99, 100

Offset: 1

John W. Layman, Mar 24 2009

The nonnegative integers with an odd number of even powers of 2 in their base-2 representation are given in A158705.
It appears that a result similar to Prouhet's theorem holds for the terms of A158704 and A158705, specifically: Sum_{k=0..2^n-2, k has an even number of even powers of 2} k^j = Sum_{k=0..2^n-2, k has an odd number of even powers of 2} k^j, for 0 <= j <= (n-1)/2. For a recent treatment of this theorem, see the reference.
Conjecture: take any binary vector of length 4n+3 with n >= 0. We can activate any bits. When a bit is activated, neighboring bits change their values 0 -> 1, 1 -> 0. Our goal is to turn the original binary vector into a vector of only ones by activating the bits. If the value of the binary vector belongs to this sequence, this is possible for a maximum of 4n+3 activations. - Mikhail Kurkov, Jun 01 2021

			The base-2 representation of 5 is 101, i.e., 5 = 2^2 + 2^0, with two even powers of 2. Thus 5 is a term of the sequence.

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
Index entries for 2-automatic sequences.

Cf. A112539 (characteristic function, up to offset), A158705 (complement).

Cf. A000069, A001969, A157971.

[ n : n in [0..150] | IsEven(&+Intseq(n, 4))]; // Vincenzo Librandi, Apr 13 2011

Select[Range[0, 100], EvenQ[Total[Take[Reverse[IntegerDigits[#, 2]], {1, -1, 2}]]] &] (* Amiram Eldar, Jan 04 2020 after Harvey P. Dale at A158705 *)

A158705 Nonnegative integers with an odd number of even powers of 2 in their base-2 representation.

Original entry on oeis.org

1, 3, 4, 6, 9, 11, 12, 14, 16, 18, 21, 23, 24, 26, 29, 31, 33, 35, 36, 38, 41, 43, 44, 46, 48, 50, 53, 55, 56, 58, 61, 63, 64, 66, 69, 71, 72, 74, 77, 79, 81, 83, 84, 86, 89, 91, 92, 94, 96, 98, 101, 103, 104, 106, 109, 111, 113, 115, 116, 118, 121, 123, 124, 126

Offset: 1

John W. Layman, Mar 26 2009

The nonnegative integers with an even number of even powers of 2 in their base-2 representation are given in A158704.
It appears that a result similar to Prouhet's theorem holds for the terms of A158704 and A158705, specifically: Sum_{k=0..2^n-1, k has an even number of even powers of 2} k^j = Sum_{k=0..2^n-1, k has an odd number of even powers of 2} k^j, for 0 <= j <= (n-1)/2. For a recent treatment of this theorem, see the reference.
Take any binary vector of length 4n+1 with n >= 0. We can activate any bits. When a bit is activated, neighboring bits change their values 0 -> 1, 1 -> 0. Our goal is to turn the original binary vector into a vector of only ones by activating the bits. If the value of the binary vector belongs to this sequence, this is possible for a maximum of 4n+1 activations. - Mikhail Kurkov, Jun 03 2021 [verification needed]

			The base-2 representation of 6 is 110, i.e., 6 = 2^2 + 2^1, with one even power of 2. Thus 6 is a term of the sequence.

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.
Eric Weisstein's World of Mathematics, Prouhet-Tarry-Escott Problem
Index entries for 2-automatic sequences.

Cf. A341389 (characteristic function), A158704 (complement).

Cf. A000069, A001969, A157971.

[ n : n in [0..150] | IsOdd(&+Intseq(n, 4))]; // Vincenzo Librandi, Apr 13 2011

Select[Range[100],OddQ[Total[Take[Reverse[IntegerDigits[#,2]],{1,-1,2}]]]&] (* Harvey P. Dale, Dec 23 2012 *)

A157970 Evil twin locations: first members of pairs of consecutive evil numbers.

Original entry on oeis.org

5, 9, 17, 23, 29, 33, 39, 45, 53, 57, 65, 71, 77, 85, 89, 95, 101, 105, 113, 119, 125, 129, 135, 141, 149, 153, 159, 165, 169, 177, 183, 189, 197, 201, 209, 215, 221, 225, 231, 237, 245, 249, 257, 263, 269, 277, 281, 287, 293, 297

Offset: 1

John W. Layman, Mar 10 2009

An evil number (A001969) is a nonnegative integer with an even number of ones in its binary expansion.

In the reference it is shown that these evil twins alternate with the odious twins (see A157971), which are pairs of consecutive odious numbers (A000069).

			The sequence of evil numbers (A001969) begins 0,3,5,6,9,10,12,15,17,18,20,..., so the first few evil twins are 5, 9, 17, ... .

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; also on JSTOR.
Eric Weisstein's World of Mathematics, Odious Number.

Cf. A000069, A001969, A157971, A248056.

SequencePosition[Table[If[EvenQ[DigitCount[n, 2, 1]], 1, 0], {n, 300}], {1, 1}][[All, 1]] (* Amiram Eldar, Dec 09 2019 after Harvey P. Dale at A157971 *)

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

a(n) = A248056(n) - 1. - Amiram Eldar, Jun 16 2025

A248057 Positions of 1,1 in the Thue-Morse sequence (A010060).

Original entry on oeis.org

2, 8, 14, 22, 26, 32, 38, 42, 50, 56, 62, 70, 74, 82, 88, 94, 98, 104, 110, 118, 122, 128, 134, 138, 146, 152, 158, 162, 168, 174, 182, 186, 194, 200, 206, 214, 218, 224, 230, 234, 242, 248, 254, 262, 266, 274, 280, 286, 290, 296, 302, 310, 314, 322, 328

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) = 2 and a(2) = 8.

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

Cf. A010060, A091855, A091785, A157971, A248056.

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]],{1,1}][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 16 2017 *)

t(n)=hammingweight(n)%2;
for(n=1,500,if(t(n)==1&&t(n-1)==1,print1(n,", "))); \\ Joerg Arndt, Mar 12 2022

a(n) = 2*A091855(n) for n >= 1.

a(n) = A157971(n) + 1. - Amiram Eldar, Jun 16 2025

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

A198681 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3k+1.

Original entry on oeis.org

3, 9, 24, 27, 42, 48, 60, 66, 72, 81, 96, 102, 114, 120, 126, 138, 144, 159, 168, 174, 180, 192, 198, 213, 216, 231, 237, 243, 258, 264, 276, 282, 288, 300, 306, 321, 330, 336, 342, 354, 360, 375, 378, 393, 399, 408, 414, 429, 432, 447, 453, 465, 471, 477, 492, 498

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.

Michel Marcus, Table of n, a(n) for n = 1..1001 (after Harvey P. Dale).
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, A158704, A198680, A198682.

Select[3Range[200],IntegerQ[(Total[IntegerDigits[#,3]]-1)/3]&] (* Harvey P. Dale, Feb 05 2012 *)

Offset corrected by Michel Marcus, Mar 02 2016

A198682 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3*k+2.

Original entry on oeis.org

6, 12, 18, 30, 36, 51, 54, 69, 75, 84, 90, 105, 108, 123, 129, 141, 147, 153, 162, 177, 183, 195, 201, 207, 219, 225, 240, 246, 252, 267, 270, 285, 291, 303, 309, 315, 324, 339, 345, 357, 363, 369, 381, 387, 402, 411, 417, 423, 435, 441, 456, 459, 474, 480, 486

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.

For each m, the sequence contains exactly one of 9*m, 9*m+3, 9*m+6. - Robert Israel, Mar 04 2016

Robert Israel, 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, A158704, A198680, A198681.

select(t -> convert(convert(t,base,3),`+`) mod 3 = 2, [seq(3*i,i=1..1000)]); # Robert Israel, Mar 04 2016

Select[Range[3, 498, 3], IntegerQ[(-2 + Plus@@IntegerDigits[#, 3])/3] &] (* Alonso del Arte, Nov 02 2011 *)

isok(n) = !(n % 3) && !((vecsum(digits(n, 3)) - 2) % 3); \\ Michel Marcus, Mar 02 2016

Offset corrected by Michel Marcus, Mar 02 2016

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 *)

A331831 Numbers k such that k and k + 1 are both negabinary odious numbers.

Original entry on oeis.org

3, 11, 15, 23, 29, 35, 43, 47, 53, 59, 63, 71, 77, 83, 91, 95, 103, 109, 115, 119, 125, 131, 139, 143, 151, 157, 163, 171, 175, 181, 187, 191, 199, 205, 211, 215, 221, 227, 235, 239, 245, 251, 255, 263, 269, 275, 283, 287, 295, 301, 307, 311, 317, 323, 331, 335

Offset: 1

Amiram Eldar, Jan 28 2020

			3 is a term since both 3 and 3 + 1 = 4 are negabinary odious numbers (A268273): 3 has 3 digits of 1 in its negabinary representation, 111, 4 has 1 digit of 1 in its negabinary representation, 100, and both 3 and 1 are odd.

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

Cf. A157971, A268273, A331830.

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

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A158704 Nonnegative integers with an even number of even powers of 2 in their base-2 representation.

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Magma

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A158705 Nonnegative integers with an odd number of even powers of 2 in their base-2 representation.

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Magma

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A157970 Evil twin locations: first members of pairs of consecutive evil numbers.

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A248057 Positions of 1,1 in the Thue-Morse sequence (A010060).

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

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A198681 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3k+1.

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A198682 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3*k+2.

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