A327495 -id:A327495 - cp's OEIS frontend

A327496 a(n) = a(n - 1) * 4^r where r = valuation(n, 2) if 4 divides n else r = (n mod 2) + 1; a(0) = 1. The denominators of A327495.

1, 16, 64, 1024, 16384, 262144, 1048576, 16777216, 1073741824, 17179869184, 68719476736, 1099511627776, 17592186044416, 281474976710656, 1125899906842624, 18014398509481984, 4611686018427387904, 73786976294838206464, 295147905179352825856, 4722366482869645213696

Offset: 0

Peter Luschny, Sep 29 2019

Cf. A327491, A327492, A327493, A327494, A327495.

Cf. A056982.

A327496 := n -> denom(add(j!^2 / (2^j*iquo(j, 2)!)^4, j=0..n)):
seq(A327496(n), n=0..19);

a(n) = 1 << (4*n - 2*hammingweight(n>>1)); \\ Kevin Ryde, May 31 2022

@cached_function
def A327496(n):
    if n == 0: return 1
    r = valuation(n, 2) if 4.divides(n) else n % 2 + 1
    return 4^r * A327496(n-1)
print([A327496(n) for n in (0..19)])

a(n) = denominator(r(n)) where r(n) = Sum_{j=0..n} j!^2 / (2^j*floor(j/2))^4.

a(n) = 4^A327492(n). - Kevin Ryde, May 31 2022

A327492 Partial sums of A327491.

Original entry on oeis.org

0, 2, 3, 5, 7, 9, 10, 12, 15, 17, 18, 20, 22, 24, 25, 27, 31, 33, 34, 36, 38, 40, 41, 43, 46, 48, 49, 51, 53, 55, 56, 58, 63, 65, 66, 68, 70, 72, 73, 75, 78, 80, 81, 83, 85, 87, 88, 90, 94, 96, 97, 99, 101, 103, 104, 106, 109, 111, 112, 114, 116, 118, 119

Offset: 0

Peter Luschny, Sep 27 2019

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A327491, A005187, A327493, A327494, A327495.

bitcount(n) = sum(digits(n, base = 2))
A327492(n) = 2n - bitcount(n) + mod(n, 2)
[A327492(n) for n in 0:62] |> println # Peter Luschny, Oct 03 2019

# For len >= 1:
A327492_list := len -> ListTools:-PartialSums([seq(A327491(j), j=0..len-1)]):
A327492_list(99)

a[n_] := 2*n + Mod[n, 2] - DigitCount[2*n, 2, 1]; Array[a, 100, 0] (* Amiram Eldar, Aug 30 2024 *)

seq(n)={my(a=vector(n+1)); for(n=1, n, a[n+1] = a[n] + if(n%4, n%2 + 1, valuation(n,2))); a} \\ Andrew Howroyd, Sep 28 2019

a(n) = n<<1 - hammingweight(n) + bittest(n,0); \\ Kevin Ryde, May 31 2022

@cached_function
def A327492(n):
    if n == 0: return 0
    r = valuation(n, 2) if 4.divides(n) else n % 2 + 1
    return r + A327492(n-1)
print([A327492(n) for n in (0..19)])

a(n) = A005187(n) + n mod 2.

a(n) ~ 2*n. - Amiram Eldar, Aug 30 2024

A327491 a(0) = 0. If 4 divides n then a(n) = valuation(n, 2) else a(n) = (n mod 2) + 1.

Original entry on oeis.org

0, 2, 1, 2, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 1, 2, 5, 2, 1, 2, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 1, 2, 6, 2, 1, 2, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2

Offset: 0

Peter Luschny, Sep 27 2019

nonn

			Seen as an irregular table for n >= 1:
  2,
  1, 2,
  2, 2, 1, 2,
  3, 2, 1, 2, 2, 2, 1, 2,
  4, 2, 1, 2, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 1, 2,
  5, 2, 1, 2, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, ....

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A327492, A327493, A327494, A327495.

A327491 := n -> if n = 0 then 0
elif 0 = irem(n, 4) then padic[ordp](n, 2)
elif 0 = irem(n, 2) then 1 else 2 fi:
seq(A327491(n), n=0..87);

a[0] = 0; a[n_] := If[Divisible[n, 4], IntegerExponent[n, 2], Mod[n, 2] + 1]; Array[a, 100, 0] (* Amiram Eldar, Aug 30 2024 *)

a(n)={if(n==0, 0, if(n%4, n%2 + 1, valuation(n,2)))} \\ Andrew Howroyd, Sep 28 2019

def A327491(n):
    if n == 0: return 0
    if 4.divides(n): return valuation(n, 2)
    return n % 2 + 1
print([A327491(n) for n in (0..87)])

a(0) = 0; if n is odd then a(n) = 2, otherwise a(n) = A007814(n). - Andrey Zabolotskiy, Jan 08 2024

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Aug 30 2024

A327493 a(n) = 2^A327492(n).

Original entry on oeis.org

1, 4, 8, 32, 128, 512, 1024, 4096, 32768, 131072, 262144, 1048576, 4194304, 16777216, 33554432, 134217728, 2147483648, 8589934592, 17179869184, 68719476736, 274877906944, 1099511627776, 2199023255552, 8796093022208, 70368744177664, 281474976710656, 562949953421312

Offset: 0

Peter Luschny, Sep 27 2019

nonn

Cf. A327492, A327491, A327494, A327495, A056040.

bitcount(n) = sum(digits(n, base = 2))
A327493(n) = 2^(2n - bitcount(n) + mod(n, 2))
[A327493(n) for n in 0:26] |> println # Peter Luschny, Oct 03 2019

A327493 := n -> 2^(A327492_list(n+1)[n+1]):
seq(A327493(n), n = 0..26);

seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[n+1] = a[n] * 2^if(n%4, n%2 + 1, valuation(n,2))); a} \\ Andrew Howroyd, Sep 28 2019

a(n)={ denominator(sum(j=0, n, j!/(2^j*(j\2)!)^2)) } \\ Andrew Howroyd, Sep 28 2019

a(n) = denominator(b(n)) where b(n) = n!/(2^n*floor(n/2)!)^2 is the normalized swinging factorial (A056040).

A327494 a(n) = numerator(r(n)) where r(n) = Sum_{j=0..n} j!/(2^j*floor(j/2)!)^2.

Original entry on oeis.org

1, 5, 11, 47, 191, 779, 1563, 6287, 50331, 201639, 403341, 1614057, 6456459, 25828839, 51658107, 206638863, 3306228243, 13225022367, 26450056889, 105800458501, 423201880193, 1692808490741, 3385617069661, 13542470306761, 108339763130127, 433359069421483

Offset: 0

Peter Luschny, Sep 27 2019

			r(n) = 1, 5/4, 11/8, 47/32, 191/128, 779/512, 1563/1024, 6287/4096, 50331/32768, 201639/131072, ...

Denominators are in A327493.

Cf. A327491, A327492, A327493, A327495, A118273, A163590.

A327494(n) = sum(<<(A163590(k), A327492(n) - A327492(k)) for k in 0:n) # Peter Luschny, Oct 03 2019

A327494 := n -> numer(add(j!/(2^j*iquo(j,2)!)^2, j=0..n)):
seq(A327494(n), n=0..25);

a(n)={ numerator(sum(j=0, n, j!/(2^j*(j\2)!)^2)) } \\ Andrew Howroyd, Sep 28 2019

Lim_{n -> oo} r(n) = (4/3)^(3/2) = A118273.

cp's OEIS Frontend

A327496 a(n) = a(n - 1) * 4^r where r = valuation(n, 2) if 4 divides n else r = (n mod 2) + 1; a(0) = 1. The denominators of A327495.

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A327492 Partial sums of A327491.

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A327491 a(0) = 0. If 4 divides n then a(n) = valuation(n, 2) else a(n) = (n mod 2) + 1.

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A327493 a(n) = 2^A327492(n).

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A327494 a(n) = numerator(r(n)) where r(n) = Sum_{j=0..n} j!/(2^j*floor(j/2)!)^2.

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