A331739 -id:A331739 - cp's OEIS frontend

A153733 Remove all trailing 1's in the binary representation of n.

0, 0, 2, 0, 4, 2, 6, 0, 8, 4, 10, 2, 12, 6, 14, 0, 16, 8, 18, 4, 20, 10, 22, 2, 24, 12, 26, 6, 28, 14, 30, 0, 32, 16, 34, 8, 36, 18, 38, 4, 40, 20, 42, 10, 44, 22, 46, 2, 48, 24, 50, 12, 52, 26, 54, 6, 56, 28, 58, 14, 60, 30, 62, 0, 64, 32, 66, 16, 68, 34, 70, 8, 72, 36, 74, 18, 76, 38

Offset: 0

Reinhard Zumkeller, Dec 31 2008

a(n) is also the map n -> A065423(n+1) applied A007814(n+1) times. - Federico Provvedi, Dec 14 2021

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to binary expansion of n

Cf. A000265, A007814, A007088.
Cf. A163575.
Cf. A065423, A331739.

a153733 n = if b == 0 then n else a153733 n'  where (n', b) = divMod n 2
-- Reinhard Zumkeller, Jul 22 2014

f:= n -> (n+1)/2^padic:-ordp(n+1,2)-1:
map(f, [$0..100]); # Robert Israel, Mar 18 2018

Table[If[EvenQ[n],n,FromDigits[Flatten[Most[Split[IntegerDigits[n,2]]]],2]],{n,0,100}] (* Harvey P. Dale, Feb 15 2014 *)
a[n_] := BitShiftRight[n + 1, IntegerExponent[n+1, 2]] - 1; a[Range[0,100]] (* Federico Provvedi, Dec 21 2021 *)

A153733(n)=(n+=1)>>valuation(n,2)-1 \\ most efficient variant: use this. - M. F. Hasler, Mar 16 2018

{a(n)=while(bittest(n,0),n>>=1);n} \\ for illustration: as long as there's a trailing bit 1, remove it. - M. F. Hasler, Mar 16 2018

a(n)=for(i=0,n,bittest(n,i)||return(n>>i)) \\ scan the trailing 1's, then remove all of them at once. - M. F. Hasler, Mar 16 2018

def a(n):
    while n&1: n >>= 1
    return n
print([a(n) for n in range(100)]) # Michael S. Branicky, Dec 18 2021

def A153733(n): return n>>(~(n+1)&n).bit_length() # Chai Wah Wu, Jul 08 2022

a(n) = n if n is even, a((n-1)/2) if odd.
a(n)/2 = A025480(n).
a(n) = A000265(n+1) - 1. - M. F. Hasler, Mar 16 2018
a(n) = n - A331739(n+1). - Federico Provvedi, Dec 21 2021

A331738 Multiplicative with a(p^e) = p^(e-A000265(e)), where A000265(x) gives the odd part of x.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Feb 02 2020

Antti Karttunen, Table of n, a(n) for n = 1..20736
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A331737, A331739.

f[p_, e_] := p^(e - e/2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 24 2022 *)

A000265(n) = (n>>valuation(n,2));
A331738(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^(f[k, 2]-A000265(f[k, 2]))); };

Multiplicative with a(p^e) = p^A331739(e).

a(n) = n / A331737(n).

A334070 Number of even-order elements in the multiplicative group of integers modulo n.

Original entry on oeis.org

0, 0, 1, 1, 3, 1, 3, 3, 3, 3, 5, 3, 9, 3, 7, 7, 15, 3, 9, 7, 9, 5, 11, 7, 15, 9, 9, 9, 21, 7, 15, 15, 15, 15, 21, 9, 27, 9, 21, 15, 35, 9, 21, 15, 21, 11, 23, 15, 21, 15, 31, 21, 39, 9, 35, 21, 27, 21, 29, 15, 45, 15, 27, 31, 45, 15, 33, 31, 33, 21, 35, 21, 63

Offset: 1

Robert A. Jones, Apr 13 2020

nonn

The number of even-order elements in a finite abelian group G is |G| - b(|G|), where b is given by A000265. To see this, decompose G as a product of cyclic groups of orders {m_k}. G has [prod_k b(m_k)] elements of odd order, since an element has odd order if and only if all its components have odd order, and each C_m factor has b(m) elements of odd order. Since b can be pulled outside the product, G has b(|G|) elements of odd order. Using that the order of (Z/nZ)^x is phi(n), we obtain a(n) = phi(n) - b(phi(n)).

Since phi(n) is even when n > 2, a(n) is odd when n > 2.

			For n = 10, the elements of (Z_n)^x with even order are 3 (order 4), 7 (order 4), and 9 (order 2). Thus, a(10) = 3.

Cf. A000010, A053575, A129527, A331739 (number of even-order elements in Z_n).

a:= n-> (t-> t-t/2^padic[ordp](t, 2))(numtheory[phi](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 14 2020

a[n_] := Length@
  Select[Range[n] - 1, EvenQ[MultiplicativeOrder[#, n]] &];
oddPart[n_] := n/2^IntegerExponent[n,2];
a[n_] := EulerPhi[n] - oddPart[EulerPhi[n]];

a(n) = A000010(n) - A053575(n) = A331739(A000010(n)).

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A153733 Remove all trailing 1's in the binary representation of n.

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A331738 Multiplicative with a(p^e) = p^(e-A000265(e)), where A000265(x) gives the odd part of x.

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Mathematica

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A334070 Number of even-order elements in the multiplicative group of integers modulo n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula