A214682 Remove 2's that do not contribute to a factor of 4 from the prime factorization of n.

1, 1, 3, 4, 5, 3, 7, 4, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 20, 21, 11, 23, 12, 25, 13, 27, 28, 29, 15, 31, 16, 33, 17, 35, 36, 37, 19, 39, 20, 41, 21, 43, 44, 45, 23, 47, 48, 49, 25, 51, 52, 53, 27, 55, 28, 57, 29

Offset

1,3

Comments

In this sequence, the number 4 exhibits some characteristics of a prime number since all extraneous 2's have been removed from the prime factorizations of all other numbers.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n) = (n*4^(v_4(n)))/(2^(v_2(n))) where v_k(n) is the k-adic valuation of n. That is, v_k(n) is the largest power of k, a, such that k^a divides n.

For n odd, a(n)=n since n has no factors of 2 (or 4).

From Peter Munn, Nov 29 2020: (Start)

a(A003159(n)) = n.

a(A036554(n)) = n/2.

a(n) = n/A056832(n) = n/A059895(n, 2) = min(n, A073675(n)).

a(A059897(n, k)) = A059897(a(n), a(k)).

(End)

Multiplicative with a(2^e) = 2^(2*floor(e/2)), and a(p^e) = p^e for odd primes p. - Amiram Eldar, Dec 09 2020

Sum_{k=1..n} a(k) ~ (5/12) * n^2. - Amiram Eldar, Nov 10 2022

Dirichlet g.f.: zeta(s-1)*(2^s+1)/(2^s+2). - Amiram Eldar, Dec 30 2022

Example

For n=8, v_4(8)=1, v_2(8)=3, so a(8)=(8*4^1)/(2^3)=4.
For n=12, v_4(12)=1, v_2(12)=2, so a(12)=(12*4^1)/(2^2)=12.

Mathematica

a[n_] := n/(2^Mod[IntegerExponent[n, 2], 2]); Array[a, 100] (* Amiram Eldar, Dec 09 2020 *)

Prog

(Sage)
C = []
for i in [1..n]:
    C.append(i*(4^(Integer(i).valuation(4))/(2^(Integer(i).valuation(2)))
(PARI) a(n)=n>>(valuation(n, 2)%2) \\ Charles R Greathouse IV, Jul 26 2012
(Python)
def A214682(n): return n>>1 if (~n&n-1).bit_length()&1 else n # Chai Wah Wu, Jan 09 2023

Crossrefs

Cf. A214681, A214685.

Range of values: A003159.

Missing values: A036554.

A056832, A059895, A073675 are used in a formula defining this sequence.

A059897 is used to express relationship between terms of this sequence.

Cf. A007814 (v_2(n)), A235127 (v_4(n)).

Sequence in context: A354932 A123901 A349164 * A093395 A176774 A126352

Adjacent sequences: A214679 A214680 A214681 * A214683 A214684 A214685

Keyword

easy,nonn,mult

Author

Tyler Ball, Jul 25 2012

Status

approved