A102391 -id:A102391 - cp's OEIS frontend

A102393 A wicked evil sequence.

1, 0, 0, 4, 0, 6, 7, 0, 0, 10, 11, 0, 13, 0, 0, 16, 0, 18, 19, 0, 21, 0, 0, 24, 25, 0, 0, 28, 0, 30, 31, 0, 0, 34, 35, 0, 37, 0, 0, 40, 41, 0, 0, 44, 0, 46, 47, 0, 49, 0, 0, 52, 0, 54, 55, 0, 0, 58, 59, 0, 61, 0, 0, 64, 0, 66, 67, 0, 69, 0, 0, 72, 73, 0, 0, 76, 0, 78, 79, 0, 81, 0, 0, 84, 0, 86

Offset: 0

Paul Barry, Jan 06 2005

Elements of A026147 (evil numbers plus one) appear at positions indexed by the evil numbers A001969, 0 otherwise. A000027(n) = A102393(n) + A102394(n).

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

Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.

Cf. A000027, A000120, A001969, A010060, A026147, A102391, A102393, A102394.

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

a[n_] := If[EvenQ @ DigitCount[n, 2, 1], n + 1, 0]; Array[a, 100, 0] (* Amiram Eldar, Aug 02 2020 *)

a(n) = (n+1)(1+(-1)^A000120(n))/2.

a(n) = (n+1)(1+(-1)^A010060(n))/2.

A102392 Odious numbers in odious places.

Original entry on oeis.org

0, 1, 2, 0, 4, 0, 0, 7, 8, 0, 0, 11, 0, 13, 14, 0, 16, 0, 0, 19, 0, 21, 22, 0, 0, 25, 26, 0, 28, 0, 0, 31, 32, 0, 0, 35, 0, 37, 38, 0, 0, 41, 42, 0, 44, 0, 0, 47, 0, 49, 50, 0, 52, 0, 0, 55, 56, 0, 0, 59, 0, 61, 62, 0, 64, 0, 0, 67, 0, 69, 70, 0, 0, 73, 74, 0, 76, 0, 0, 79, 0, 81, 82, 0, 84, 0

Offset: 0

Paul Barry, Jan 06 2005

Odious numbers A000069(n) appear at positions indexed by the odious numbers, 0 otherwise.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000069, A000120, A001477, A010060, A102391, A102392.

odnQ[n_]:=OddQ[DigitCount[n,2,1]]; Table[If[odnQ[n],n,0],{n,0,100}] (* Harvey P. Dale, Oct 15 2019 *)

def A102392(n): return n*(n.bit_count()&1) # Chai Wah Wu, Nov 23 2023

A001477(n) = A102391(n)+a(n).
a(n) = if((1+floor(n/2))(1+(-1)^A000120(n))/2=0, n, 0).
a(n) = if((1+floor(n/2))(1+(-1)^A010060(n))/2=0, n, 0).
a(n) = n*A010060(n). - Ridouane Oudra, Apr 19 2025

A367513 The exponentially evil part of n: the largest unitary divisor of n that is an exponentially evil number (A262675).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Nov 21 2023

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

Cf. A001969, A010059, A034444, A102391, A262675, A270428, A366902, A366904, A367512, A367516.

Similar sequences: A350388, A350389, A366906, A367168, A367514.

f[p_, e_] := p^(e * (1 - ThueMorse[e])); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(hammingweight(f[i, 2])%2, 1, f[i, 1]^f[i, 2]));}

from math import prod
from sympy import factorint
def A367513(n): return prod(p**e for p, e in factorint(n).items() if e.bit_count()&1^1) # Chai Wah Wu, Nov 23 2023

Multiplicative with a(p^e) = p^(e*A010059(e)) = p^A102391(e).
a(n) = n/A367514(n).
A001221(a(n)) = A367512(n).
A034444(a(n)) = A367516(n).
a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
a(n) <= n, with equality if and only if n is an exponentially evil number (A262675).

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A102393 A wicked evil sequence.

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A102392 Odious numbers in odious places.

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A367513 The exponentially evil part of n: the largest unitary divisor of n that is an exponentially evil number (A262675).

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