A131323 -id:A131323 - cp's OEIS frontend

A161817 Positions n such that A010060(n) = A010060(n+5).

0, 2, 5, 8, 10, 11, 12, 14, 15, 16, 18, 21, 24, 26, 29, 32, 34, 37, 40, 42, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 62, 63, 64, 66, 69, 72, 74, 75, 76, 78, 79, 80, 82, 85, 88, 90, 93, 96, 98, 101, 104, 106, 107, 108, 110, 111, 112, 114, 117, 120, 122, 125, 128, 130, 133, 136, 138, 139, 140, 142, 143, 144

Offset: 1

Vladimir Shevelev, Jun 20 2009

Let A=Axxxxxx be any sequence. Denote by A^* the intersection of A and the union of sequences {4*A(n)+k}, k=-1,0,1,2. Then the present sequence is the union of A079523^* and A121539^*.

Conjecture. In every sequence of numbers n such that A010060(n)=A010060(n+k) for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. [Vladimir Shevelev, Jul 31 2009]

G. C. Greubel, Table of n, a(n) for n = 1..10000
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv:1401.3727 [math>NT], 2014.
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
V. Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence, arXiv:0907.0880 [math.NT], 2009-2012.

Cf. A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n - 1)/2]; Reap[For[n = 0, n <= 20000, n++,  If[tm[n] == tm[n + 5], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)

is(n)=hammingweight(n+5)==Mod(hammingweight(n),2) \\ Charles R Greathouse IV, Mar 26 2013

A161824 Numbers such that A010060(n) = A010060(n+6).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 12, 13, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 44, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 61, 64, 65, 66, 67, 70, 71, 72, 73, 76, 77, 80, 81, 82, 83, 86, 87, 88, 89, 90, 91, 94, 95, 96, 97, 98, 99, 102, 103, 104, 105, 108

Offset: 1

Vladimir Shevelev, Jun 20 2009

Let A=Axxxxxx be any sequence from OEIS. Denote by A^* the intersection of the union of sequences {2*A(n)+j}, j=0,1, and the union of sequences {4*A(n)+k}, k=-2,-1,0,1. Then the sequence is the union of (A079523)^* and (A121539)^*.

G. C. Greubel, Table of n, a(n) for n = 1..10000
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv:1401.3727 [math.NT], 2014.
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
Vladimir Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence, arXiv:0907.0880 [math.NT], 2009-2012.

Cf. A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n - 1)/2]; Reap[For[n = 0, n <= 6000, n++, If[tm[n] == tm[n + 6], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)

is(n)=hammingweight(n)%2==hammingweight(n+6)%2 \\ Charles R Greathouse IV, Aug 20 2013

Terms a(40) onwards added by G. C. Greubel, Jan 05 2018

Offset corrected by Mohammed Yaseen, Mar 29 2023

A161890 Numbers such that A010060(n) = A010060(n+9).

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 9, 13, 15, 16, 18, 19, 20, 22, 24, 26, 27, 28, 30, 32, 34, 35, 36, 38, 39, 41, 45, 47, 48, 50, 51, 52, 54, 55, 57, 61, 63, 64, 66, 67, 68, 70, 71, 73, 77, 79, 80, 82, 83, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 103, 105, 109, 111, 112, 114, 115, 116, 118, 120

Offset: 0

Vladimir Shevelev, Jun 21 2009

Or union of intersection of A161639 and {A079523(n)-8} and intersection of A161673 and {A121539(n)-8}. In general, for a>=1, consider equations A010060(x+a)+A010060(x)=1, A010060(x+a)=A010060(x). Denote via B_a (C_a) the sequence of nonnegative solutions of the first (second) equation. Then we have recursions: B_(a+1) is the union of transactions 1) C_a and {A121539(n)-a}, 2) B_a and {A079523(n)-a}; C_(a+1) is the union of transactions 1) C_a and {A079523(n)-a}, 2) B_a and {A121539(n)-a}.
Conjecture. In every sequence of numbers n, such that A010060(n)=A010060(n+k), for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. - Vladimir Shevelev, Jul 31 2009
This conjecture was actually proved in a later version of the Shevelev arxiv article cited below, and it can also easily be proved by the Walnut prover. - Jeffrey Shallit, Oct 12 2022

G. C. Greubel, Table of n, a(n) for n = 0..10000
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv:1401.3727 [math.NT], 2014.
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
Vladimir Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence, arXiv:0907.0880[math.NT], 2009-2012.

Cf. A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n - 1)/2]; Reap[For[n = 0, n <= 18000, n++, If[tm[n] == tm[n + 9], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)
SequencePosition[ThueMorse[Range[0,150]],{x_,,,_,,,_,,,x_}][[All,1]]-1 (* Harvey P. Dale, Feb 06 2023 *)

is(n)=hammingweight(n)%2==hammingweight(n+9)%2 \\ Charles R Greathouse IV, Aug 20 2013

Terms a(35) onward added by G. C. Greubel, Jan 05 2018

A225822 Lesser of adjacent odd numbers with different parity of binary weight and both isolated from odd numbers of same parity of binary weight.

Original entry on oeis.org

7, 23, 31, 39, 55, 71, 87, 95, 103, 119, 127, 135, 151, 159, 167, 183, 199, 215, 223, 231, 247, 263, 279, 287, 295, 311, 327, 343, 351, 359, 375, 383, 391, 407, 415, 423, 439, 455, 471, 479, 487, 503, 511, 519, 535, 543, 551, 567, 583, 599, 607, 615, 631

Offset: 1

Brad Clardy, Jul 30 2013

Write the sequence of odious odd numbers above the sequence of evil odd numbers connecting all that are 2 apart:
1 7 11-13 19-21 25 31 35-37 41 47-49 55 59-61 67-69 73 79-81 87 91-93 97
3-5  9 15-17 23 27-29 33 39 43-45 51-53 57 63-65 71 75-77 83-85 89 95 99-
Remove the connected numbers:
1 7             25 31       41       55             73       87       97
     9       23       33 39             57       71             89  95
Define these as "isolated".
The sequence is the smaller of the remaining pairs that are 2 apart.
The 1 is not a member since there is no change in parity between 1 and 7.
All of the differences between adjacent numbers in both the evil and odious sequences are either 2, 4 or 6, with 4 being the indicator that a transition in parity occurs. The program provided utilizes that fact to produce the sequence.
The sequence that includes all numbers along this path is A047522 (numbers congruent to {1,7} mod 8). This is also the same as the odd terms of A199398 (XOR of the first n odd numbers).
This sequence is similar to A044449 (numbers n such that string 1,3 occurs in the base 4 representation of n but not of n+1), but it contains additional terms. An example is 119. Its base 4 representation is 1313 while the base 4 representation of 120 is 1320. It may be that another workable definition of the sequence is -- numbers n in base 4 representation such that string 1,3 occurs one less time in n+1 than n, but I have not been able to check this.
The difference between the numbers in the sequence is always either 8 or 16, however there appears to be no recurring repetitions in it. Writing the 8 as a 0 and the 16 as a 1 (or dividing the difference pattern by 2 and subtracting a 1) produces a difference pattern of: 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1... which is an infinite word.
A similar process writing Even Odious over Even Evils produces 6, 22, 30, 38, 54, 70... which is twice A131323 (Odd numbers n such that the binary expansion ends in an even number of 1's), with all numbers along the path given by A047451 (numbers congruent to  {0,6} mod 8) and yields the same difference pattern which produces the same infinite word.

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A001969 (evil numbers), A129771 (odd evil numbers).
Cf. A000069 (odious numbers), A092246 (odd odious numbers).
Cf. A047522 (numbers congruent to {1,7} mod 8).
Cf. A199398 (XOR of first n odd numbers).
Cf. A044449 (a subset of this sequence).
Cf. A131323 (odd numbers n such that the binary expansion ends in an even number of 1's).
Cf. A047451 (numbers congruent to {0,6} mod 8).
Cf. A000120 (binary weight of n).
Cf. A079523.

//Function Bweight calculates the binary weight of an integer
Bweight := function(m)
Bweight:=0;
adigs := Intseq(m,2);
for n:= 1 to Ilog2(m)+1 do
  Bweight:=Bweight+adigs[n];
end for;
return Bweight;
end function;
prevodi:=0;
currodi:=0;
m:=0;
count:=0;
for n:= 1 to 20000 by 2 do
    m:=m+1;
    if (Bweight(n) mod 2 eq 1) then odious:=true; else odious:=false; end if;
    if (odious) then currodi:=n; end if;
       if (currodi - prevodi eq 4) then
          if (m mod 2 eq 1) then count:=count+1; count,n-2;
             else count:=count+1;count,n-4;
       end if;
    end if;
    if(odious) then prevodi:=currodi; end if;
end for;

2*Select[Range[1, 320, 2], EvenQ[IntegerExponent[# + 1, 2]] &] + 1 (* Amiram Eldar, Jul 24 2023 *)

is(n)=n%4==3 && valuation(n\4+1, 2)%2 \\ Charles R Greathouse IV, Aug 20 2013

from itertools import count, islice
def A225822_gen(startvalue=1): # generator of terms >= startvalue
    return map(lambda m:(m<<1)+1,filter(lambda n:n&1 and not (~(n+1)&n).bit_length()&1,count(max(startvalue,1))))
A225822__list = list(islice(A225822_gen(),30)) # Chai Wah Wu, Jul 09 2022

a(n) = 2*A131323(n) + 1.
a(n) = 4*A079523(n) + 3. - Charles R Greathouse IV, Aug 20 2013
a(n) ~ 12n. (In particular, a(n) = 12n + O(log n).) - Charles R Greathouse IV, Aug 20 2013

A233388 Odious numbers that are the sum of 2 consecutive odious numbers.

Original entry on oeis.org

11, 19, 35, 47, 59, 67, 79, 91, 107, 115, 131, 143, 155, 171, 179, 191, 203, 211, 227, 239, 251, 259, 271, 283, 299, 307, 319, 331, 339, 355, 367, 379, 395, 403, 419, 431, 443, 451, 463, 475, 491, 499, 515, 527, 539, 555, 563, 575, 587, 595

Offset: 1

Gerasimov Sergey, Dec 08 2013

Composites in this sequence: 35, 91, 115, 143, 155, 171, 203, 259, 299, 319, 229, 366, 395, 403, 451, 475, 515, 527, 539, 555, 575, 595,...

(a(n + 1) - a(n))/4 are 2, 4, 3, 3, 2, 3, 3, 4, 2, 4, 3, 3, 4, 2, 3, 3, 2, 4, 3, 3, 2, 3, 3, 4, 2, 3, 3, 2, 4, 3, 3, 4, 2, 4, 3, 3, 2, 3,...

			11 is in this sequence because 11 = A000069(6) and 11 = 4 + 7 where 4 = A000069(3) and 7 = A000069(4).
19 is in this sequence because 19 = A000069(10) and 19 = 8 + 11 where 8 = A000069(5) and 11 = A000069(6).

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

Cf. A000069, A227930, A131323, A234431.

odQ[n_] := OddQ @ DigitCount[n, 2, 1]; Select[Plus @@@ Partition[Select[Range[300] , odQ], 2, 1], odQ] (* Amiram Eldar, Aug 29 2020 *)

A234431 Numbers that are the sum of 2 successive evil numbers (A001969).

Original entry on oeis.org

3, 8, 11, 15, 19, 22, 27, 32, 35, 38, 43, 47, 51, 56, 59, 63, 67, 70, 75, 79, 83, 88, 91, 94, 99, 104, 107, 111, 115, 118, 123, 128, 131, 134, 139, 143, 147, 152, 155, 158, 163, 168, 171, 175, 179, 182, 187, 191, 195, 200, 203, 207, 211, 214, 219, 224, 227, 230, 235, 239, 243, 248, 251

Offset: 1

Irina Gerasimova, Dec 26 2013

First differences are in {3, 4, 5}; 4*n - 2 <= a(n) <= 4*n. - Charles R Greathouse IV, Dec 26 2013

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

Cf. A001969, A003159 (indices of odd numbers in A234431), A131323 (odd numbers in A234431).

Total/@Partition[Select[Range[0,200],EvenQ[DigitCount[#,2,1]]&],2,1] (* Harvey P. Dale, Nov 02 2015 *)

a(n)=4*n+hammingweight(n-1)%2+hammingweight(n)%2-2 \\ Charles R Greathouse IV, Dec 26 2013

a(n) = A001969(n) + A001969(n + 1).

A234011 The sums of 2 consecutive odious numbers (A000069).

Original entry on oeis.org

3, 6, 11, 15, 19, 24, 27, 30, 35, 40, 43, 47, 51, 54, 59, 63, 67, 72, 75, 79, 83, 86, 91, 96, 99, 102, 107, 111, 115, 120, 123, 126, 131, 136, 139, 143, 147, 150, 155, 160, 163, 166, 171, 175, 179, 184, 187, 191, 195, 198, 203, 207, 211, 216, 219, 222, 227, 232, 235, 239, 243, 246

Offset: 1

Gerasimov Sergey, Dec 27 2013

The union of A131323(k) and (A225822(m)+(-1)^m).
All even numbers in this sequence are evil numbers (A001969).
It seems that A233388(n) = a(A091785(n)).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000069, A003159 (indices of odd numbers in A234011), A036554 (indices of even numbers in A234011), A131323 (odd sums of 2 successive odious or 2 successive evil numbers), A233388 (odious numbers in A234011), A234431 (sums of 2 consecutive evil numbers), A017101, A091785, A225822, A227930, A233388.

Plus @@@ Partition[Select[Range[125], OddQ[DigitCount[#, 2][[1]]] &], 2, 1] (* Amiram Eldar, Jul 24 2023 *)

a(n)=4*n-hammingweight(n-1)%2-hammingweight(n)%2 \\ Charles R Greathouse IV, Dec 29 2013

(define (A234011 n) (+ (A000069 n) (A000069 (+ n 1)))) ;; Antti Karttunen, Dec 29 2013

a(n) = A000069(n) + A000069(n + 1).
4n - 2 <= a(n) <= 4n. - Charles R Greathouse IV, Dec 29 2013
a(2n+1) = 8n + 3 = A017101(n). - Ralf Stephan, Dec 31 2013

Terms recomputed and checked by Antti Karttunen, Dec 29 2013

A161974 a(n) = number of equalities of the form A010060(n+k) = A010060(n), k=1,2,3.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1

Offset: 0

Vladimir Shevelev, Jun 23 2009

See comment to A161916. 3-a(n) is the number of equalities of kind A010060(n+k) = 1-A010060(n), k=1,2,3.

Cf. A161916, A161890, A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060.

a(n)=sum(k=1,3,hammingweight(n)%2==hammingweight(n+k)%2) \\ Charles R Greathouse IV, Aug 20 2013

Missing a(24)=1 inserted by Georg Fischer, Jun 21 2024

A162311 Numbers such that A010060(n) = A010060(n+7).

Original entry on oeis.org

1, 3, 4, 5, 7, 10, 14, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 42, 46, 49, 51, 52, 53, 55, 58, 62

Offset: 1

Vladimir Shevelev, Jul 01 2009

Or union of intersection of A161673 and {A121539(n)-7} and intersection of A161639 and {A079523(n)-7}.

Conjecture: In every sequence of numbers n, such that A010060(n)=A010060(n+k), for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. - Vladimir Shevelev, Jul 31 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv:1401.3727 [math.NT], 2014.
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
V. Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence, arXiv:0907.0880 [math.NT], 2009-2012.

Cf. A161890, A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706, A161916, A161974.

tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n - 1)/2]; Reap[For[n = 0, n <= 20000, n++, If[tm[n] == tm[n + 7], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)

is(n)=hammingweight(n)%2==hammingweight(n+7)%2 \\ Charles R Greathouse IV, Aug 20 2013

A162648 Locations of patterns 1001 or 0110 in the Thue-Morse sequence A010060.

Original entry on oeis.org

0, 4, 6, 8, 12, 16, 20, 22, 24, 28, 30, 32, 36, 38, 40, 44, 48, 52, 54, 56, 60, 64, 68, 70, 72, 76, 80, 84, 86, 88, 92, 94, 96, 100, 102, 104, 108, 112, 116, 118, 120, 124, 126, 128, 132, 134, 136, 140, 144, 148, 150, 152, 156, 158, 160, 164, 166, 168, 172, 176, 180

Offset: 1

Vladimir Shevelev, Jul 08 2009

nonn

Numbers n for which A010060(n+1) = A010060(n+2) = 1-A010060(n) and A010060(n+3) = A010060(n).

Or intersection of A121539, A161674, and A161579.

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

Cf. A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

Select[Range[3500], MatchQ[IntegerDigits[#, 2], {b : (1) ..} | {_, 0, b : (1) ..} /; OddQ[Length[{b}]]] &] - 1 (* G. C. Greubel, Jan 05 2018 *)
With[{nn=200},Sort[Join[SequencePosition[ThueMorse[Range[0,nn]],{1,0,0,1}],SequencePosition[ ThueMorse[Range[0,nn]],{0,1,1,0}]]][[;;,1]]]-1 (* Harvey P. Dale, Aug 20 2024 *)

is(n)=my(v=vector(4,i,hammingweight(n+i-1))); v[1]==v[4] && v[1]!=v[2] && v[1]!=v[3] \\ Charles R Greathouse IV, Aug 20 2013

a(n) = A079523(n) - 1.

More readable definition from R. J. Mathar, Sep 16 2009

cp's OEIS Frontend

A161817 Positions n such that A010060(n) = A010060(n+5).

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A161824 Numbers such that A010060(n) = A010060(n+6).

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A161890 Numbers such that A010060(n) = A010060(n+9).

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A225822 Lesser of adjacent odd numbers with different parity of binary weight and both isolated from odd numbers of same parity of binary weight.

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A233388 Odious numbers that are the sum of 2 consecutive odious numbers.

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A234431 Numbers that are the sum of 2 successive evil numbers (A001969).

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A234011 The sums of 2 consecutive odious numbers (A000069).

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