A091072 -id:A091072 - cp's OEIS frontend

A339597 When 2*n+1 first appears in A086799.

1, 2, 5, 4, 9, 10, 13, 8, 17, 18, 21, 20, 25, 26, 29, 16, 33, 34, 37, 36, 41, 42, 45, 40, 49, 50, 53, 52, 57, 58, 61, 32, 65, 66, 69, 68, 73, 74, 77, 72, 81, 82, 85, 84, 89, 90, 93, 80, 97, 98, 101, 100, 105, 106, 109, 104, 113, 114, 117, 116, 121, 122, 125, 64, 129, 130, 133, 132, 137

Offset: 0

Marc LeBrun and N. J. A. Sloane, Jan 06 2021

Michel Marcus, Table of n, a(n) for n = 0..10000
Christian Krause, LODA program

Cf. A086799, A091072 (terms sorted), A129760.

N := 127: # for a(0) to a(N)
V := Array(0..N): count := 0:
for i from 1 while count < N+1 do
  with(MmaTranslator[Mma]):
  f(i) := BitOr(i,i-1);
  v := (f(i)-1)/2;
  if v <= N and V[v] = 0 then count := count+1; V[v] := i fi
od:
convert(V,list); # Robert Israel, Jan 07 2021

f(n) = bitor(n, n-1); \\ A086799
a(n) = my(k=1); while (f(k) != 2*n+1, k++); k; \\ Michel Marcus, Jan 07 2021

a(n) = n++; n<<1 - 1<Kevin Ryde, Mar 29 2021

def A339597(n): return ((m:=n+1)<<1)-(m&-m) # Chai Wah Wu, Sep 01 2023

a(n) = 2*(n+1) - A006519(n+1) = n+1 with a 0 bit inserted above its least significant 1-bit. - Kevin Ryde, Mar 29 2021

a(n) = A129760(n+1) + n+1. - Christian Krause, May 05 2021

A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

Original entry on oeis.org

9, 18, 25, 36, 49, 50, 72, 81, 98, 100, 121, 144, 162, 169, 196, 200, 225, 242, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1058, 1089, 1152, 1156, 1225, 1250, 1296, 1352, 1369

Offset: 1

Benoit Cloitre, Apr 18 2002

Previous name: sum(d|n,6d/(2+mu(d))) is odd, where mu(.) is the Moebius function, A008683.
From Peter Munn, Jul 06 2020: (Start)
Numbers that have an odd number of odd nonsquarefree divisors.
[Proof of equivalence to the name, where m denotes a positive integer:
(1) These properties are equivalent: (a) m has an even number of odd squarefree divisors; (b) m has a nontrivial odd part.
(2) These properties are equivalent: (a) m has an odd number of odd divisors; (b) the odd part of m is square.
(3) m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are both true or both false.
(4) The trivial odd part, 1, is a square, so (1)(b) and (2)(b) cannot both be false, which (from (1), (2)) means (1)(a) and (2)(a) cannot both be false.
(5) From (3), (4), m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are true.
(6) m satisfies the condition in the name if and only if (1)(b) and (2)(b) are true, which (from (1), (2)) is equivalent to (1)(a) and (2)(a) being true, and hence from (5), to m satisfying the condition at the start of this comment.]
(End)
Numbers whose sum of non-unitary divisors (A048146) is odd. - Amiram Eldar, Sep 16 2024

			To determine the odd part of 18, remove all factors of 2, leaving 9. 9 is a nontrivial square, so 18 is in the sequence. - _Peter Munn_, Jul 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Odd part.

A000265, A008683 are used in definitions of this sequence.
Lists of numbers whose odd part satisfies other conditions: A028982 (square), A028983 (nonsquare), A029747 (less than 6), A029750 (less than 8), A036349 (even number of prime factors), A038550 (prime), A070776 U {1} (power of a prime), A072502 (square of a prime), A091067 (has form 4k+3), A091072 (has form 4k+1), A093641 (noncomposite), A105441 (composite), A116451 (greater than 4), A116882 (less than or equal to even part), A116883 (greater than or equal to even part), A122132 (squarefree), A229829 (7-rough), A236206 (11-rough), A260488\{0} (has form 6k+1), A325359 (proper prime power), A335657 (odd number of prime factors), A336101 (prime power).
Cf. A048146, A091476.

Select[Range[1000], (odd = #/2^IntegerExponent[#, 2]) > 1 && IntegerQ @ Sqrt[odd] &] (* Amiram Eldar, Sep 29 2020 *)

upto(n) = { my(res = List()); forstep(i = 3, sqrtint(n), 2, for(j = 0, logint(n\i^2, 2), listput(res, i^2<David A. Corneth, Sep 28 2020

Sum_{n>=1} 1/a(n) = 2 * Sum_{k>=1} 1/(2*k+1)^2 = Pi^2/4 - 2 = A091476 - 2 = 0.467401... - Amiram Eldar, Feb 18 2021

New name from Peter Munn, Jul 06 2020

A106841 Numbers m such that m, m+1 and m+2 have odd part of the form 4*k+1.

Original entry on oeis.org

8, 16, 32, 40, 64, 72, 80, 104, 128, 136, 144, 160, 168, 200, 208, 232, 256, 264, 272, 288, 296, 320, 328, 336, 360, 392, 400, 416, 424, 456, 464, 488, 512, 520, 528, 544, 552, 576, 584, 592, 616, 640, 648, 656, 672, 680, 712, 720, 744, 776, 784, 800, 808

Offset: 1

Ralf Stephan, May 03 2005

nonn

Either of form 2a(m) or 32k + 8, k >= 0, 0 < m < n.
Number points of the Heighway/Harter dragon curve starting m=0 at the origin.  Those m with odd part 4k+1 (A091072) are where the curve turns left.  So this sequence is the first m of each run of 3 consecutive left turns.  There are no runs of 4 or more since the turn at odd m alternates left and right.  Bates, Bunder, and Tognetti (theorem 19 page 104), show this sequence is integers of the form 2^p*(4k+1) with p>=3.  From which a(n) = 8*A091072(n) as Ralf Stephan already noted. - Kevin Ryde, Jan 28 2020
The asymptotic density of this sequence is 1/16. - Amiram Eldar, Sep 14 2024

			40/8 = 5 is 1 mod 4 and so is 41 and 42/2 = 21, thus 40 is in sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Bruce Bates, Martin Bunder, and Keith Tognetti, Mirroring and Interleaving in the Paperfolding Sequence, Applicable Analysis and Discrete Mathematics, Volume 4, Number 1, April 2010, pages 96-118.

Cf. A091072, A106840, A106838.

Equals 8 * A091072.

opn[n_]:=n/2^IntegerExponent[n,2]; Transpose[Select[Partition[Range[ 1000],3,1],Mod[opn/@#,4]=={1,1,1}&]][[1]] (* Harvey P. Dale, May 15 2011 *)

lista(nn) = for(k=1, nn, if(((k/2^valuation(k, 2)-1)/2)%2==0, print1(8*k, ", "))); \\ Jinyuan Wang, Jan 30 2020

A119972 a(n) = n * A034947(n).

Original entry on oeis.org

1, 2, -3, 4, 5, -6, -7, 8, 9, 10, -11, -12, 13, -14, -15, 16, 17, 18, -19, 20, 21, -22, -23, -24, 25, 26, -27, -28, 29, -30, -31, 32, 33, 34, -35, 36, 37, -38, -39, 40, 41, 42, -43, -44, 45, -46, -47, -48, 49, 50, -51, 52, 53, -54, -55, -56, 57, 58, -59, -60, 61, -62, -63, 64, 65, 66, -67, 68, 69, -70, -71, 72, 73, 74, -75

Offset: 1

Alford Arnold, Jun 01 2006

Previous name was: Flag n when the first difference of the decimal encoding of the Gray code is negative. (With "flag" meaning negate n when the difference is negative.)

Merge A091072 with minus A091067 maintaining increasing absolute value.

			A003188 begins 0  1  3  2  6  7  5  4 12  13  15  14  10  11  9 ... so
A055975 begins   1  2 -1  4  1 -2 -1  8  1   2  -1  -4   1  -2  ...
Sequence         1, 2,-3, 4, 5,-6,-7, 8, 9, 10,-11,-12, 13,-14, ...
Negative terms are at positions 3,6,7,11,12,14,..., = A091067.
Positive terms are the complement, which is A091072.

Cf. A034947, A003188, A055975, A091067, A091072.

isA091067 := proc(n) option remember ; if n mod 4 = 3 then RETURN(true) ; else if n mod 2 = 0 then if isA091067(n/2) then RETURN(true) ; fi ; fi ; RETURN(false) ; fi ; end: A119972 := proc(n) if isA091067(n) then -n ; else n ; fi ; end: for n from 1 to 180 do printf("%d, ",A119972(n)) ; od ; # R. J. Mathar, May 14 2007
# second Maple program:
a:= n-> numtheory[jacobi](-1, n)*n:
seq(a(n), n=1..75);  # Alois P. Heinz, Jan 19 2023

a[n_] := n KroneckerSymbol[-1, n];
Array[a, 75] (* Jean-François Alcover, Apr 09 2020 *)

a(n) = n*kronecker(-1, n); \\ Andrew Howroyd, Aug 06 2018

a(n) = n*Kronecker(-1, n) = n * A034947(n). - Andrew Howroyd, Aug 06 2018

More terms from R. J. Mathar, May 14 2007
Keyword:mult added by Andrew Howroyd, Aug 06 2018
New name using existing formula from Joerg Arndt, Jan 19 2023

A379015 a(n) is the reversed non-adjacent form (NAF) representation of n.

Original entry on oeis.org

0, 1, 1, -3, 1, 5, -3, -7, 1, 9, 5, -19, -3, 13, -7, -15, 1, 17, 9, -11, 5, 21, -19, -35, -3, 29, 13, -39, -7, 25, -15, -31, 1, 33, 17, -23, 9, 41, -11, -27, 5, 37, 21, -83, -19, 45, -35, -67, -3, 61, 29, -51, 13, 77, -39, -71, -7, 57, 25, -79, -15, 49, -31, -63

Offset: 0

Darío Clavijo, Dec 13 2024

Fixed points exist when the non-adjacent form is palindromic.

			For n=7 a(7) = -7 because:
7 to NAF encoding read from least to most significant bit: [-1, 0, 0, 1]
Reversed: [1, 0, 0, -1]
NAF to integer: -7.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Wikipedia, Non-adjacent form.

Cf. A000079, A004767, A016813, A091067, A091072, A334913.

a[n_]:=Module[{E=n,r=0},While[E>0,If[OddQ[E],Module[{Zi=2-Mod[E,4]},E-=Zi;r+=Zi;]];E=Floor[E/2];r*=2;];Floor[r/2]];Table[a[n],{n,0,63}] (* James C. McMahon, Dec 26 2024 *)

a(n) = { my (r = 0, d); while (n, if (n%2, d = 2 - (n % 4); r += d; n -= d;); r *= 2; n \= 2;); return (r \ 2); } \\ Rémy Sigrist, Dec 28 2024

def a(n):
    E, r = n, 0
    while E:
        if E & 1:
            Zi = 2 - (E & 3)
            E -= Zi
            r += Zi
        E >>= 1
        r <<= 1
    return r >> 1
print([a(n) for n in range(0,64)])

a(2^k) = 1.
a(A091072(n)) > 0 iff a(n) is in A016813.
a(A091067(n)) < 0 iff abs(a(n)) is in A004767.

a(0) = 0 prepended by Rémy Sigrist, Dec 28 2024

A088023 Set a(1) = 1. Then take the list of defined initial terms, reverse their order, add 1, 2, 3, ... to the reversed list in succession and append this new list to the right of the previously defined terms. Repeat this process indefinitely.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 8, 9, 9, 9, 10, 11, 12, 12, 13, 14, 14, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 19, 20, 20, 21, 22, 22, 22, 23, 24, 25, 25, 25, 26, 26, 26, 26, 27, 28, 28, 29, 30, 30, 30, 30, 31, 32, 32, 32, 33, 33, 33

Offset: 1

Gary W. Adamson, Sep 19 2003

nonn

Conjecture: a(n+1) >= a(n). Comments from Don Reble, Nov 13 2005: The conjecture is plainly true. In fact, a(n+1)-a(n) = 0 or 1. Also a(A091072(n)) = n; a(A091072(n)+1) = n+1.

			The sequence begins 1, 2, then reverse 1, 2 = 2, 1 then add 1, 2 to the latter getting 3, 3. Then append 3, 3, to the right of 1, 2, getting 1, 2, 3, 3. Then repeating the instructions, 1, 2, 3, 3 is reversed then add 1, 2, 3, 4 to 3, 3, 2, 1, = 4, 5, 5, 5. Append the latter to 1, 2, 3, 3 getting 1, 2, 3, 3, 4, 5, 5, 5, ...; and so on.

a(n)=2a(n/2)-1 if a=2^k else a(n)=a(2^k-n+1)+n-2^(k-1) if 2^(k-1)

Edited by John W. Layman, Oct 10 2003

A106840 Numbers m such that both m and m+1 have odd part of the form 4*k+1.

Original entry on oeis.org

1, 4, 8, 9, 16, 17, 20, 25, 32, 33, 36, 40, 41, 49, 52, 57, 64, 65, 68, 72, 73, 80, 81, 84, 89, 97, 100, 104, 105, 113, 116, 121, 128, 129, 132, 136, 137, 144, 145, 148, 153, 160, 161, 164, 168, 169, 177, 180, 185, 193, 196, 200, 201, 208, 209, 212, 217, 225

Offset: 1

Ralf Stephan, May 03 2005

nonn

From Amiram Eldar, Sep 14 2024: (Start)
Disjoint union of A017077 and {4*A091072(n)}.
The asymptotic density of this sequence is 1/4. (End)

			20/4 = 5 == 1 (mod 4) and also 21 == 1 (mod 4), therefore 20 is in the sequence.

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

Cf. A017077, A091072, A106841, A106837.

Contains A106841 and A106841+1.

f[n_] := Mod[n / 2^IntegerExponent[n, 2] - 1, 4]; SequencePosition[Array[f, 250], {0, 0}][[;;,1]] (* Amiram Eldar, Sep 14 2024 *)

A119973 Numbers of the form (4k+1)*2^j which are not a sum of two squares.

Original entry on oeis.org

21, 33, 42, 57, 66, 69, 77, 84, 93, 105, 114, 129, 132, 133, 138, 141, 154, 161, 165, 168, 177, 186, 189, 201, 209, 210, 213, 217, 228, 237, 249, 253, 258, 264, 266, 273, 276, 282, 285, 297, 301, 308, 309, 321, 322, 329, 330, 336, 341, 345, 354, 357, 372

Offset: 1

Alford Arnold, Jun 03 2006

Intersection of A091072 and A022544. - Robert Israel, Oct 28 2018

			42 is there because it's (4*5+1)*2^1 and is not a sum of two squares.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001481, A022544, A084109, A091072, A120772.

filter:= proc(n) local w; w:= n/2^padic:-ordp(n,2);
w mod 4 = 1 and select(t -> t[2]::odd and t[1] mod 4 = 3, ifactors(w)[2]) <> []
end proc:
select(filter, [$1..1000]); # Robert Israel, Oct 28 2018

okQ[n_] := EvenQ[(n/2^IntegerExponent[n, 2]-1)/2] && SquaresR[2, n] == 0;
Select[Range[1000], okQ] (* Jean-François Alcover, Feb 09 2023 *)

More terms from Don Reble, Jul 24 2006

A337821 For n >= 0, a(4n+1) = 0, a(4n+3) = a(2n+1) + 1, a(2n+2) = a(n+1).

Original entry on oeis.org

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

Offset: 1

Peter Munn, Sep 23 2020

This sequence is the ruler sequence A007814 interleaved with this sequence; specifically, the odd bisection is A007814, the even bisection is the sequence itself.
The 3-adic valuation of the Doudna sequence (A005940).
The 2-adic valuation of Kimberling's paraphrase (A003602) of the binary number system. [Edited Peter Munn, Aug 13 2025.]

			Start of table showing the interleaving with ruler sequence, A007814:
   n  a(n)  A007814    a(n/2)
            ((n+1)/2)
   1   0       0
   2   0                 0
   3   1       1
   4   0                 0
   5   0       0
   6   1                 1
   7   2       2
   8   0                 0
   9   0       0
  10   0                 0
  11   1       1
  12   1                 1
  13   0       0
  14   2                 2
  15   3       3
  16   0                 0
  17   0       0
  18   0                 0
  19   1       1
  20   0                 0
  21   0       0
  22   1                 1
  23   2       2
  24   1                 1

Odd bisection: A007814.
A000265, A003602, A005940, A007949 are used in a formula defining this sequence.
Positions of zeros: A091072.
Sequences with similar interleaving: A089309, A014577, A025480, A034947, A038189, A082392, A099545, A181363, A274139.

a[n_] := IntegerExponent[(n/2^IntegerExponent[n, 2] + 1)/2, 2]; Array[a, 100] (* Amiram Eldar, Sep 30 2020 *)

a(n) = valuation(n>>valuation(n,2)+1, 2) - 1; \\ Kevin Ryde, Apr 06 2024

a(2*n) = a(n).
a(2*n+1) = A007814(n+1).
a(n) = A007949(A005940(n)).
a(n) = A007814(A003602(n)) = A007814((A000265(n)+1) / 2) = A089309(n) - 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Sep 13 2024

A088742 Run lengths of A088023.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 1, 2, 1, 4, 1, 3, 3, 1, 1, 2, 1, 3, 1, 1, 3, 4, 1, 2, 1, 4, 1, 3, 3, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 1, 2, 1, 4, 1, 3, 4, 1, 2, 1, 3, 1, 1, 3, 4, 1, 2, 1, 4, 1, 3, 3, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 1, 2, 1, 4, 1, 3, 3, 1, 1, 2, 1, 3, 1, 1, 3, 4, 1, 2, 1, 4, 1, 3, 4, 1

Offset: 1

Gary W. Adamson, Oct 12 2003

nonn

First differences of A091072.

Cf. A088023, A088743, A088744.

More terms and better description from Ralf Stephan, Sep 03 2004

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A339597 When 2*n+1 first appears in A086799.

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A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

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A106841 Numbers m such that m, m+1 and m+2 have odd part of the form 4*k+1.

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A119972 a(n) = n * A034947(n).

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A379015 a(n) is the reversed non-adjacent form (NAF) representation of n.

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A088023 Set a(1) = 1. Then take the list of defined initial terms, reverse their order, add 1, 2, 3, ... to the reversed list in succession and append this new list to the right of the previously defined terms. Repeat this process indefinitely.

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A106840 Numbers m such that both m and m+1 have odd part of the form 4*k+1.

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