A125592 -id:A125592 - cp's OEIS frontend

A092246 Odd "odious" numbers (A000069).

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, 103, 107, 109, 115, 117, 121, 127, 131, 133, 137, 143, 145, 151, 155, 157, 161, 167, 171, 173, 179, 181, 185, 191, 193, 199, 203, 205, 211, 213, 217, 223, 227, 229, 233

Offset: 1

Benoit Cloitre, Feb 23 2004

In other words, numbers having a binary representation ending in 1, and an odd number of 1's overall. It follows that by decrementing an odd odious number, one gets an even evil number (A125592). - Ralf Stephan, Aug 27 2013
The members of the sequence may be called primitive odious numbers because every odious number is a power of 2 times one of these numbers. Note that the difference between consecutive terms is either 2, 4, or 6. - T. D. Noe, Jun 06 2007
From Gary W. Adamson, Apr 06 2010: (Start)
a(n) = A026147(n)-th odd number, where A026147 = (1, 4, 6, 7, 10, 11, ...); e.g.,
         n: 1   2   3   4   5   6   7   8   9  10  11
  n-th odd: 1   3   5   7   9  11  13  15  17  19  21
      a(n): 1           7      11  13          19  21
etc. (End)
Numbers m, such that when merge-sorting lists of length m, the maximal number of comparisons is even: A003071(a(n)) = A230720(n). - Reinhard Zumkeller, Oct 28 2013
Fixed points of permutation pair A268717/A268718. - Antti Karttunen, Feb 29 2016

Cf. A129771, A026147.
Cf. A230709 (complement).
Cf. A268717, A268718, A268673.

a092246 n = a092246_list !! (n - 1)
a092246_list = filter odd a000069_list
-- Reinhard Zumkeller, Oct 28 2013

Table[If[n < 1, 0, 2 n - 1 - Mod[First@ DigitCount[n - 1, 2], 2]], {n, 120}] /. n_ /; EvenQ@ n -> Nothing (* Michael De Vlieger, Feb 29 2016 *)
Select[Range[1, 1001, 2], OddQ[Total[IntegerDigits[#, 2]]]&] (* Jean-François Alcover, Mar 15 2016 *)

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

a(n)=4*n-if(hammingweight(n-1)%2,1,3) \\ Charles R Greathouse IV, Mar 22 2013

def A092246(n): return (n<<2)-(1 if (n-1).bit_count()&1 else 3) # Chai Wah Wu, Mar 03 2023

a(n) = 4*n + 2*A010060(n-1) - 3;

a(n) = 2*A001969(n-1) + 1.

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

Original entry on oeis.org

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

A367803 Exponentially evil squares.

Original entry on oeis.org

1, 64, 729, 1024, 4096, 15625, 46656, 59049, 117649, 262144, 531441, 746496, 1000000, 1048576, 1771561, 2985984, 3779136, 4826809, 7529536, 9765625, 11390625, 16000000, 16777216, 24137569, 34012224, 47045881, 60466176, 64000000, 85766121, 113379904, 120472576, 148035889

Offset: 1

Amiram Eldar, Dec 01 2023

Numbers whose prime factorization contains only exponents that are even evil numbers (A125592).
Also, squares of exponentially evil numbers (A262675).
Also, numbers with an equal number of exponentially odious and exponentially evil divisors, i.e., numbers k such that A366901(k) = A366902(k). - Amiram Eldar, Feb 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.

Intersection of A000290 and A262675.
Cf. A366901, A366902, A367801, A367802, A367804.
Cf. A001969, A125592.

evilQ[n_] := EvenQ[DigitCount[n, 2, 1]]; Select[Range[10^4]^2, #== 1 || AllTrue[FactorInteger[#][[;;, 2]], evilQ] &]

isexpevil(n) = {my(f = factor(n)); for (i = 1, #f~, if(hammingweight(f[i, 2])%2, return (0))); 1;}
is(n) = issquare(n) && isexpevil(n);

a(n) = A262675(n)^2.

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^A125592(k)) = Product_{p prime} f(1/p) = 1.01833932269003592136..., where f(x) = (2/(1-x^2) + Product_{k>=0} (1 - x^(2^k)) + Product_{k>=0} (1 - (-x)^(2^k)))/4.

A375850 The maximum even exponent in the prime factorization of n!, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 4, 2, 4, 8, 8, 10, 10, 2, 6, 6, 6, 16, 16, 18, 18, 4, 4, 22, 22, 10, 6, 6, 6, 26, 26, 14, 4, 32, 32, 34, 34, 8, 18, 38, 38, 6, 6, 6, 10, 42, 42, 46, 46, 22, 12, 12, 12, 50, 50, 26, 4, 54, 54, 56, 56, 28, 30, 30, 30, 64, 64, 66, 66, 32, 32, 70

Offset: 0

Amiram Eldar, Aug 31 2024

The sequence of indices of record values, 0, 6, 10, 12, 18, 20, 24, 30, 34, 36, 40, ..., are the evil numbers (A001969) multiplied by 2 (A125592).

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers.

Cf. A000142, A001969, A011371, A125592, A375033, A375849.

a[n_] := Max[0, Max[Select[FactorInteger[n!][[;; , 2]], EvenQ]]]; Array[a, 100, 0]

a(n) = {my(e = select(x -> !(x % 2), factor(n!)[, 2])); if(#e == 0, 0, vecmax(e));}

from collections import Counter
from sympy import factorint
def A375850(n): return max(filter(lambda x: x&1^1,sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values()),default=0) # Chai Wah Wu, Aug 31 2024

a(n) = A375033(n!).

max(a(n), A375849(n)) = A011371(n).

A084682 Even evil numbers with an even digital sum.

Original entry on oeis.org

0, 6, 20, 24, 40, 46, 48, 60, 66, 68, 80, 86, 114, 116, 130, 132, 136, 150, 154, 156, 170, 172, 178, 190, 192, 198, 202, 204, 222, 226, 228, 240, 246, 260, 264, 282, 284, 288, 312, 318, 330, 332, 338, 350, 354, 356, 374, 378, 390, 394, 396, 402, 404, 408, 420

Offset: 1

Jason Earls, Jun 30 2003

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

Cf. A001969.

Intersection of A054683 and A125592.

filter:= n -> convert(convert(n,base,2),`+`)::even and convert(convert(n,base,10),`+`)::even:
select(filter, [seq(i,i=2..10000,2)]); # Robert Israel, Dec 31 2024

eee[n_] :=  And @@ EvenQ /@ {n, Count[IntegerDigits[n, 2], 1], Total[IntegerDigits[n]]};
Select[Range[0, 420], eee] (* Jake L Lande, Jun 30 2024 *)

is(n)={ bitand(n,1)==0 && bitand(sumdigits(n),1)==0 && bitand(hammingweight(n),1)==0 }
select(is, [0..500]) \\ Joerg Arndt, Jun 30 2024

Offset changed by Andrew Howroyd, Sep 18 2024

A348416 For n >= 1; a(n) = gcd(n,w(n)) where w(n) is the binary weight of n, A000120(n).

Original entry on oeis.org

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

Offset: 1

Ctibor O. Zizka, Oct 19 2021

a(n) is even for n in A125592, a(n) = 1 for n in A094387.

			n = 6; gcd(6, A000120(6)) = 2, thus a(6) = 2.

Cf. A000120, A001969, A049445, A094387, A125592.

Array[GCD[#, Total@ IntegerDigits[#, 2]] &, 105] (* Michael De Vlieger, Oct 19 2021 *)

a(n) = gcd(n, hammingweight(n)); \\ Michel Marcus, Oct 19 2021

a(n) = gcd(n, A000120(n)).

a(n) = A000120(n) if and only if n is in A049445. - Amiram Eldar, Oct 19 2021

cp's OEIS Frontend

A092246 Odd "odious" numbers (A000069).

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

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A367803 Exponentially evil squares.

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A375850 The maximum even exponent in the prime factorization of n!, or 0 if no such exponent exists.

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A084682 Even evil numbers with an even digital sum.

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A348416 For n >= 1; a(n) = gcd(n,w(n)) where w(n) is the binary weight of n, A000120(n).

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