A327493 -id:A327493 - cp's OEIS frontend

A327492 Partial sums of A327491.

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

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.

A327495 a(n) = numerator( Sum_{j=0..n} (j!/(2^j*floor(j/2)!)^2)^2 ).

Original entry on oeis.org

1, 17, 69, 1113, 17817, 285297, 1141213, 18260633, 1168681737, 18699007017, 74796032037, 1196736992841, 19147791938817, 306364680039081, 1225458720340365, 19607339566855065, 5019478929156305865, 80311662878468159865, 321246651514020383485, 5139946424277661728785

Offset: 0

Peter Luschny, Sep 27 2019

This sequence is a variant of the Landau constants when the normalized central binomial is replaced by the normalized swinging factorial.
(1) A277233(n)/4^A005187(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2 with the normalized central binomial
    g(n) = (2*n)! / (2^n*n!)^2 = A001790(n)/A046161(n).
(2) A327495(n)/4^A327492(n) are the rationals considered here. These numbers are defined as H(n) = Sum_{j=0..n} h(j)^2 with the normalized swinging factorial
    h(n) = n! / (2^n*floor(n/2)!)^2 = A163590(n)/A327493(n).
(3) In particular, this means that we have the pure integer representations
    A277233(n) = Sum_{k=0..n}(A001790(k)*(2^(A005187(n) - A005187(k))))^2;
    A327495(n) = Sum_{k=0..n}(A163590(k)*(2^(A327492(n) - A327492(k))))^2.
(4) A163590 is the odd part of the swinging factorial and A001790 is the odd part of the swinging factorial at even indices (see a comment from Aug 01 2009 in A001790). Similarly, A327493(2n)=A046161(2n) and A327493(2n+1) = 2*A046161(2n+1).
(5) A005187 are the partial sums of A001511, the 2-adic valuation of 2n, and A327492 are the partial sums of A327491.

			r(n) = 1, 17/16, 69/64, 1113/1024, 17817/16384, 285297/262144, 1141213/1048576, 18260633/16777216, ...

Denominators in A327496.
Cf. A327491, A327492, A327493, A327494.
Cf. A277233, A005187, A056040, A000984, A001790/A046161, A163590, A001511.

A327495 := n -> numer(add(j!^2/(2^j*iquo(j,2)!)^4, j=0..n)):
seq(A327495(n), n=0..19);

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

Denominator(r(n)) = 4^A327492(n) = A327493(n)^2 = A327496(n).

a(n) = Sum_{k=0..n} (A163590(k)*(2^(A327492(n) - A327492(k))))^2.

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.

Original entry on oeis.org

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

cp's OEIS Frontend

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

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A327495 a(n) = numerator( Sum_{j=0..n} (j!/(2^j*floor(j/2)!)^2)^2 ).

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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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