A092246 -id:A092246 - cp's OEIS frontend

A228495 Characteristic function of the odd odious numbers (A092246).

0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 0

Ralf Stephan, Aug 23 2013

The following sequences all appear to have the same parity: A003071, A029886, A061297, A092524, A093431, A102393, A104258, A122248, A128975. - Jeremy Gardiner, Dec 28 2008.

a(n+1) is the characteristic function of the even evil numbers (A125592). - Jeremy Gardiner, Feb 06 2015

Antti Karttunen, Table of n, a(n) for n = 0..65537
Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.
Index entries for characteristic functions.
Index entries for sequences related to binary expansion of n.

Cf. A000035, A010060, A092246, A092436, A125592, A254379.

Cf. A003071, A029886, A061297, A092524, A093431, A102393, A104258, A122248, A128975.

a[n_] := If[OddQ[n] && OddQ[DigitCount[n, 2, 1]], 1, 0]; Array[a, 100, 0] (* Amiram Eldar, Aug 06 2023 *)

a(n)=if(n%2==0,0,subst(Pol(binary((n-1)/2)),x,1)%2==0)

A228495(n) = ((n%2)&&(hammingweight(n)%2)); \\ Antti Karttunen, Jan 12 2019

def A228495(n): return n.bit_count()&1&n # Chai Wah Wu, Mar 03 2023

a(2n) = 0, a(2n+1) = A092436(n).

a(n) = A000035(n) * A010060(n). - Antti Karttunen, Jan 12 2019

A268673 a(0) = 0; a(1) = 1; for n > 1, a(n) = 1 + 4*A092246(n-1).

Original entry on oeis.org

0, 1, 5, 29, 45, 53, 77, 85, 101, 125, 141, 149, 165, 189, 197, 221, 237, 245, 269, 277, 293, 317, 325, 349, 365, 373, 389, 413, 429, 437, 461, 469, 485, 509, 525, 533, 549, 573, 581, 605, 621, 629, 645, 669, 685, 693, 717, 725, 741, 765, 773, 797, 813, 821, 845, 853, 869, 893, 909, 917, 933, 957, 965, 989, 1005

Offset: 0

Antti Karttunen, Feb 19 2016

nonn

Seems to be also the fixed points of permutations A268823 and A268824.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A268823, A268824.
Cf. also A092246, A268717.
Many (but not all) terms of A013710 seem to be included.

Join[{0, 1}, 1 + 4 Select[Range[1, 251, 2], OddQ[Total[IntegerDigits[#, 2]]]&]] (* Jean-François Alcover, Mar 15 2016 *)

def A268673(n): return (((m:=n-2)<<4)+(13 if m.bit_count()&1 else 5)) if n>1 else n # Chai Wah Wu, Mar 03 2023

(define (A268673 n) (if (<= n 1) n (+ 1 (* 4 (A092246 (- n 1))))))

a(0) = 0; a(1) = 1; for n > 1, a(n) = 1 + 4*A092246(n-1).

A254614 Union of odd odious (A092246) and evil (A001969) numbers.

Original entry on oeis.org

0, 1, 3, 5, 6, 7, 9, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 43, 45, 46, 47, 48, 49, 51, 53, 54, 55, 57, 58, 59, 60, 61, 63, 65, 66, 67, 68, 69, 71, 72, 73, 75, 77, 78, 79, 80, 81, 83, 85, 86, 87, 89

Offset: 1

Jeremy Gardiner, Feb 03 2015

Complement of A128309 (Even odious numbers).

			A128309 begins 2,4,8, hence this sequence begins 0,1,3,5,6,7.

Cf. A001969, A092246, A128309, A230709.

Select[Range[0, 100], OddQ[#] || !OddQ[DigitCount[#, 2, 1]] &] (* Amiram Eldar, Aug 07 2023 *)

A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.

Original entry on oeis.org

2, 3, 7, 11, 13, 19, 25, 31, 37, 41, 47, 55, 59, 61, 67, 73, 87, 91, 97, 103, 109, 115, 117, 131, 137, 143, 145, 157, 167, 171, 185, 191, 193, 203, 211, 213, 229, 239, 241, 247, 253, 283, 285, 299, 301, 313, 319, 333, 351, 355, 357, 361, 369, 375

Offset: 1

David Petry (petry(AT)accessone.com)

nonn

Or, binary irreducible polynomials, interpreted as binary vectors, then written in base 10.
The numbers {a(n)} are a subset of the set {A206074}. - Thomas Ordowski, Feb 21 2014
2^n - 1 is a term if and only if n = 2 or n is a prime and 2 is a primitive root modulo n. - Jianing Song, May 10 2021
For odd k, k is a term if and only if binary_reverse(k) = A145341((k+1)/2) is. - Joerg Arndt and Jianing Song, May 10 2021

			x^4 + x^3 + 1 -> 16+8+1 = 25. Or, x^4 + x^3 + 1 -> 11001 (binary) = 25 (decimal).

A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (first 1377 terms from T. D. Noe)
Index entries for sequences operating on GF(2)[X]-polynomials

Written in binary: A058943.
Number of degree-n irreducible polynomials: A001037, see also A000031.
Multiplication table: A048720.
Characteristic function: A091225. Inverse: A091227. a(n) = A091202(A000040(n)). Almost complement of A091242. Union of A091206 & A091214 and also of A091250 & A091252. First differences: A091223. Apart from a(1) and a(2), a subsequence of A092246 and hence A000069.
Table of irreducible factors of n: A256170.
Irreducible polynomials satisfying particular conditions: A071642, A132447, A132449, A132453, A162570.
Factorization sentinel: A278239.
Sequences analyzing the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the corresponding integer: A234741, A234742, A235032, A235033, A235034, A235035, A235040, A236850, A325386, A325559, A325560, A325563, A325641, A325642, A325643.
Factorization-preserving isomorphisms: A091203, A091204, A235041, A235042.
See A115871 for sequences related to cross-domain congruences.
Functions based on the irreducibles: A305421, A305422.

fQ[n_] := Block[{ply = Plus @@ (Reverse@ IntegerDigits[n, 2] x^Range[0, Floor@ Log2@ n])}, ply == Factor[ply, Modulus -> 2] && n != 2^Floor@ Log2@ n]; fQ[2] = True; Select[ Range@ 378, fQ] (* Robert G. Wilson v, Aug 12 2011 *)
Reap[Do[If[IrreduciblePolynomialQ[IntegerDigits[n, 2] . x^Reverse[Range[0, Floor[Log[2, n]]]], Modulus -> 2], Sow[n]], {n, 2, 1000}]][[2, 1]] (* Jean-François Alcover, Nov 21 2016 *)

is(n)=polisirreducible(Pol(binary(n))*Mod(1,2)) \\ Charles R Greathouse IV, Mar 22 2013

A027697 Odious primes: primes with odd number of 1's in binary expansion.

Original entry on oeis.org

2, 7, 11, 13, 19, 31, 37, 41, 47, 59, 61, 67, 73, 79, 97, 103, 107, 109, 127, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443, 457, 463, 487, 491, 499, 521, 541, 557, 563

Offset: 1

N. J. A. Sloane

Conjecture: a(n) < A027699(n) except for n = 2; verified up to n=5*10^7. Moreover, I conjecture that A027699(n) - a(n) tends to infinity. - Vladimir Shevelev

T. D. Noe, Table of n, a(n) for n=1..10000
E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, (French) [Sums of digits and almost primes] Math. Ann. 305 (1996), no. 3, 571--599. MR1397437 (97k:11029)
Ben Green, Three topics in additive prime number theory, arXiv:0710.0823 [math.NT], Oct 03, 2007, pp. 12-27.
Vladimir Shevelev, Generalized Newman phenomena and digit conjectures on primes, Internat. J. of Mathematics and Math. Sciences, 2008 (2008), Article ID 908045, 1-12.

Cf. A027699, A066148, A066149, A081092.

Cf. A000069 (odious numbers), A092246 (odd odious numbers)

a:=proc(n) local nn: nn:= convert(ithprime(n), base, 2): if `mod`(sum(nn[j], j =1..nops(nn)), 2)=1 then ithprime(n) else end if end proc: seq(a(n),n=1..103); # Emeric Deutsch, Oct 24 2007

Clear[BinSumOddQ];BinSumOddQ[a_]:=Module[{i,s=0},s=0;For[i=1,i<=Length[IntegerDigits[a,2]],s+=Extract[IntegerDigits[a,2],i];i++ ];OddQ[s]]; lst={};Do[p=Prime[n];If[BinSumOddQ[p],AppendTo[lst,p]],{n,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 06 2009 *)
Select[Prime@ Range@ 120, OddQ@ First@ DigitCount[#, 2] &] (* Michael De Vlieger, Feb 08 2016 *)

f(p)={v=binary(p);s=0;for(k=1,#v,if(v[k]==1,s++));return(s%2)};
forprime(p=2, 563, if(f(p), print1(p,", "))) \\ Washington Bomfim, Jan 14 2011

s=[]; forprime(p=2, 1000, if(norml2(binary(p))%2==1, s=concat(s, p))); s \\ Colin Barker, Feb 18 2014

from sympy import primerange
print([n for n in primerange(1, 1001) if bin(n)[2:].count("1")%2]) # Indranil Ghosh, May 03 2017

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A003071 Sorting numbers: maximal number of comparisons for sorting n elements by list merging.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 14, 17, 25, 27, 30, 33, 38, 41, 45, 49, 65, 67, 70, 73, 78, 81, 85, 89, 98, 101, 105, 109, 115, 119, 124, 129, 161, 163, 166, 169, 174, 177, 181, 185, 194, 197, 201, 205, 211, 215, 220, 225, 242, 245, 249, 253, 259, 263, 268, 273, 283, 287, 292, 297, 304

Offset: 1

N. J. A. Sloane

The following sequences all appear to have the same parity: A003071, A029886, A061297, A092524, A093431, A102393, A104258, A122248, A128975. - Jeremy Gardiner, Dec 28 2008

a(A092246(n)) = A230720(n); a(A230709(n)) = A230721(n+1). - Reinhard Zumkeller, Oct 28 2013

D. E. Knuth, Art of Computer Programming, Vol. 3, Sections 5.2.4 and 5.3.1.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
An Vinh Nguyen Dinh, Nhien Pham Hoang Bao, Terrillon Jean-Christophe, Hiroyuki Iida, Reaper Tournament System, 2018.
An Vinh Nguyen Dinh, Nhien Pham Hoang Bao, Mohd Nor Akmal Khalid, Hiroyuki Iida, Simulating competitiveness and precision in a tournament structure: a reaper tournament system, Int'l J. of Information Technology (2020) Vol. 12, 1-18.
Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.
Index entries for sequences related to sorting

Cf. A001855, A133457.

a003071 n = 1 - 2 ^ last es +
   sum (zipWith (*) (zipWith (+) es [0..]) (map (2 ^) es))
   where es = reverse $ a133457_row n
-- Reinhard Zumkeller, Oct 28 2013

a[1] = 0; a[n_] := a[n] = a[n-1] + 2^IntegerExponent[n-1, 2] + DigitCount[n-1, 2, 1] - 1; Table[a[n], {n, 1, 61}] (* Jean-François Alcover, Feb 10 2012, after Henry Bottomley *)

Let n = 2^e_1 + 2^e_2 + ... + 2^e_t, e_1 > e_2 > ... > e_t >= 0, t >= 1. Then a(n) = 1 - 2^e_t + Sum_{k=1..t} (e_k + k - 1)*2^e_k [Knuth, Problem 14, Section 5.2.4].
a(n) = a(n-1) + A061338(n) = a(n-1) + A006519(n) + A000120(n) - 1 = n + A000337(A000523(n)) + a(n - 2^A000523(n)). a(2^k) = k*2^k + 1 = A002064(k). - Henry Bottomley, Apr 27 2001
G.f.: x/(1-x)^3 + 1/(1-x)^2*Sum(k>=1, (-1+(1-x)*2^(k-1))*x^2^k/(1-x^2^k)). - Ralf Stephan, Apr 17 2003

More terms from David W. Wilson

A268717 Permutation of natural numbers: a(0) = 0, a(n) = A003188(1+A006068(n-1)), where A003188 is binary Gray code and A006068 is its inverse.

Original entry on oeis.org

0, 1, 3, 6, 2, 12, 4, 7, 5, 24, 8, 11, 9, 13, 15, 10, 14, 48, 16, 19, 17, 21, 23, 18, 22, 25, 27, 30, 26, 20, 28, 31, 29, 96, 32, 35, 33, 37, 39, 34, 38, 41, 43, 46, 42, 36, 44, 47, 45, 49, 51, 54, 50, 60, 52, 55, 53, 40, 56, 59, 57, 61, 63, 58, 62, 192, 64, 67, 65, 69, 71, 66, 70, 73, 75, 78, 74, 68, 76, 79, 77, 81

Offset: 0

Antti Karttunen, Feb 12 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A268718.
Cf. A000523, A003188, A003987, A006068, A010060, A066194, A101080, A171977, A268718, A268726, A268727.
Row 1 and column 1 of array A268715 (without the initial zero).
Row 1 of array A268820.
Cf. A092246 (fixed points).
Cf. A268817 ("square" of this permutation).
Cf. A268821 ("shifted square"), A268823 ("shifted cube") and also A268825, A268827 and A268831 ("shifted higher powers").

A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := If[n == 0, 0, BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}]]; a[n_] := If[n == 0, 0, A003188[1 + A006068[n-1]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 23 2016 *)

A003188(n) = bitxor(n, floor(n/2));
A006068(n) = if(n<2, n, {my(m = A006068(floor(n/2))); 2*m + (n%2 + m%2)%2});
for(n=0, 100, print1(if(n<1, 0, A003188(1 + A006068(n - 1)))", ")) \\ Indranil Ghosh, Mar 31 2017

def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    m = A006068(n//2)
    return 2*m + (n%2 + m%2)%2
def a(n): return 0 if n<1 else A003188(1 + A006068(n - 1))
print([a(n) for n in range(0, 101)]) # Indranil Ghosh, Mar 31 2017

def A268717(n):
    k, m = n-1, n-1>>1
    while m > 0:
        k ^= m
        m >>= 1
    return k+1^ k+1>>1 # Chai Wah Wu, Jun 29 2022

(define (A268717 n) (if (zero? n) n (A003188 (A066194 n))))

a(n) = A003188(A066194(n)) = A003188(1+A006068(n-1)).
Other identities. For all n >= 0:
A101080(n,a(n+1)) = 1. [The Hamming distance between n and a(n+1) is always one.]
A268726(n) = A000523(A003987(n, a(n+1))). [A268726 gives the index of the toggled bit.]
From Alan Michael Gómez Calderón, May 29 2025: (Start)
a(2*n) = (2*n-1) XOR (2-A010060(n-1)) for n >= 1;
a(n) = (A268718(n-1)-1) XOR (A171977(n-1)+1) for n >= 2. (End)

A129771 Evil odd numbers.

Original entry on oeis.org

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, 101, 105, 111, 113, 119, 123, 125, 129, 135, 139, 141, 147, 149, 153, 159, 163, 165, 169, 175, 177, 183, 187, 189, 195, 197, 201, 207, 209, 215, 219, 221, 225, 231, 235

Offset: 1

Tanya Khovanova, May 16 2007

A heuristic argument suggests that, as n tends to infinity, a(n)/n converges to 4. - Stefan Steinerberger, May 17 2007
These numbers may be called primitive evil numbers because every evil number is a power of 2 multiplied by one of these numbers. Note that the difference between consecutive terms is either 2, 4, or 6. - T. D. Noe, Jun 06 2007
If m is in the sequence, then so is 2m-1 because in binary, m is x1 and 2m-1 is x01. Presumably the numbers that generate the whole sequence by application of n -> 2n-1 are the evil numbers times 4 plus 3. - Ralf Stephan, May 25 2013

Francisco J. Muñoz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Francisco J. Muñoz and Juan Carlos Nuño, Rule-based Generation of de Bruijn Sequences: Memory and Learning, arXiv:2507.09764 [cs.FL], 2025. See p. 9.

Intersection of A001969 and A005408.
Supersequence of A093688.
Cf. A092246 (odd odious numbers).
Column 2 of A277880, positions of 1's in A277808 (2's in A277822).
Cf. A000069, A128309, A277823, A277902.

Select[Range[300], OddQ[ # ] && EvenQ[DigitCount[ #, 2, 1]] &] (* Stefan Steinerberger, May 17 2007 *)
Select[Range[300], EvenQ[Plus @@ IntegerDigits[ #, 2]] && OddQ[ # ] &]

is(n)=n%2 && hammingweight(n)%2==0 \\ Charles R Greathouse IV, Mar 21 2013

a(n)=4*n-if(hammingweight(n-1)%2,3,1) \\ Charles R Greathouse IV, Mar 21 2013

def A129771(n): return (((m:=n-1)<<1)+(m.bit_count()&1^1)<<1)+1 # Chai Wah Wu, Mar 09 2023

a(n) = 2*A000069(n) + 1. a(n) is 1 plus twice odious numbers.
a(n) = A128309(n) + 1. a(n) is 1 plus odious even numbers.
A132680(a(n)) = A132680((a(n)-1)/2) + 2. - Reinhard Zumkeller, Aug 26 2007
a(n) = 4n + O(1). - Charles R Greathouse IV, Mar 21 2013
a(n) = A001969(1+A000069(n)) = A277902(A277823(n)). - Antti Karttunen, Nov 05 2016

More terms from Stefan Steinerberger, May 17 2007

A268718 Permutation of natural numbers: a(0) = 0, a(n) = 1 + A003188(A006068(n)-1), where A003188 is binary Gray code and A006068 is its inverse.

Original entry on oeis.org

0, 1, 4, 2, 6, 8, 3, 7, 10, 12, 15, 11, 5, 13, 16, 14, 18, 20, 23, 19, 29, 21, 24, 22, 9, 25, 28, 26, 30, 32, 27, 31, 34, 36, 39, 35, 45, 37, 40, 38, 57, 41, 44, 42, 46, 48, 43, 47, 17, 49, 52, 50, 54, 56, 51, 55, 58, 60, 63, 59, 53, 61, 64, 62, 66, 68, 71, 67, 77, 69, 72, 70, 89, 73, 76, 74, 78, 80, 75, 79, 113, 81

Offset: 0

Antti Karttunen, Feb 12 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A268717.
Cf. A003188, A006068, A010060, A105081, A128309, A166519, A171977, A277822.
Row 1 of array A268830.
Cf. A092246 (fixed points).
Cf. A268818 ("square" of this permutation).
Cf. A268822 ("shifted square"), A268824 ("shifted cube") and also A268826, A268828 and A268832 (higher "shifted powers").

{0}~Join~Table[1 + BitXor[#, Floor[#/2]] &[BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}] - 1], {n, 81}] (* Michael De Vlieger, Feb 29 2016, after Jean-François Alcover at A006068 and Robert G. Wilson v at A003188 *)

a003188(n)=bitxor(n, n>>1);
a006068(n)= {
    my( s=1, ns );
    while ( 1,
        ns = n >> s;
        if ( 0==ns, break() );
        n = bitxor(n, ns);
        s <<= 1;
    );
    return (n);
} \\ by Joerg Arndt
a(n)=if(n==0, 0, 1 + a003188(a006068(n) - 1)); \\ Indranil Ghosh, Jun 07 2017

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a(n): return 0 if n==0 else 1 + a003188(a006068(n) - 1) # Indranil Ghosh, Jun 07 2017

(define (A268718 n) (if (zero? n) n (A105081 (A006068 n))))

a(0) = 0, and for n >= 1, a(n) = A105081(A006068(n)) = 1 + A003188(A006068(n)-1).
Other identities. For all n >= 1:
a(A128309(n)) = A128309(n)+2. [Maps any even odious number to that number + 2.]
From Alan Michael Gómez Calderón, May 29 2025: (Start)
a(n) - 1 = A268717(n+1) XOR (A171977(n)+1) for n >= 1;
a(2*n-1) - 1 = (2-A010060(n-1)) XOR (A166519(n-1)-1) for n >= 1;
a(2*n) - 1 = (a(2*(n+1)-1)-1) XOR 2^A277822(n) for n >= 1. (End)

A230709 Union of even odious (cf. A128309) and evil numbers (cf. A001969).

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 42, 43, 44, 45, 46, 48, 50, 51, 52, 53, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 70, 71, 72, 74, 75, 76, 77, 78, 80, 82, 83, 84, 85, 86, 88

Offset: 1

Reinhard Zumkeller, Oct 28 2013

Apart from initial zero: numbers m, such that when mergesorting lists of length m, the maximal number of comparisons is odd: A003071(a(n+1)) = A230721(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sorting

Cf. A092246 (complement).

a230709 n = a230709_list !! (n-1)
a230709_list = filter (\x -> a010060 x * x `mod` 2 == 0) [0..]

Complement[Range[0, m = 88], Select[Range[1, m, 2], OddQ[ Total[ IntegerDigits[#, 2]]]&]] (* Jean-François Alcover, Dec 15 2018 *)

is(n)=if(hammingweight(n)%2,n%2==0,1) \\ Charles R Greathouse IV, May 09 2016

A010060(a(n)) * a(n) mod 2 = 0.

cp's OEIS Frontend

A228495 Characteristic function of the odd odious numbers (A092246).

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A268673 a(0) = 0; a(1) = 1; for n > 1, a(n) = 1 + 4*A092246(n-1).

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A254614 Union of odd odious (A092246) and evil (A001969) numbers.

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A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.

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A027697 Odious primes: primes with odd number of 1's in binary expansion.

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PARI

PARI

Python

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A003071 Sorting numbers: maximal number of comparisons for sorting n elements by list merging.

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Haskell

Mathematica

Formula

Extensions

A268717 Permutation of natural numbers: a(0) = 0, a(n) = A003188(1+A006068(n-1)), where A003188 is binary Gray code and A006068 is its inverse.

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