A249153 -id:A249153 - cp's OEIS frontend

A069895 2^a(n) divides (2n)^(2n): exponent of 2 in (2n)^(2n).

2, 8, 6, 24, 10, 24, 14, 64, 18, 40, 22, 72, 26, 56, 30, 160, 34, 72, 38, 120, 42, 88, 46, 192, 50, 104, 54, 168, 58, 120, 62, 384, 66, 136, 70, 216, 74, 152, 78, 320, 82, 168, 86, 264, 90, 184, 94, 480, 98, 200, 102, 312, 106, 216, 110, 448, 114, 232, 118, 360, 122

Offset: 1

Labos Elemer, Apr 10 2002

Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.

Cf. A001511, A007814, A085534, A091512, A249153 (partial sums).

function A069895List(length)
    a = zeros(Int, length)
    for n in 1:length a[n] = 2 * (isodd(n) ? n : n + a[div(n, 2)]) end
a end
A069895List(61) |> println # Peter Luschny, Oct 16 2021

a:= 2*n*padic[ordp](2*n, 2):
seq(a(n), n=1..61);  # Alois P. Heinz, Oct 14 2021

Table[ Part[ Flatten[ FactorInteger[n^n]], 2], {n, 2, 124, 2}]

a(n) = n<<=1; n*valuation(n,2); \\ Kevin Ryde, Oct 14 2021

def A069895(n): return n*(n&-n).bit_length()<<1 # Chai Wah Wu, Jul 11 2022

a(n) = 2*n*A001511(n).
a(n) = A007814(A085534(n)). [corrected by Kevin Ryde, Oct 15 2021]
G.f.: Sum_{k>=0} 2^(k+1)*x^2^k/(1-x^2^k)^2. - Ralf Stephan, Jun 07 2003
a(n) = 2 * A091512(n). - Alois P. Heinz, Oct 14 2021
Sum_{k=1..n} a(k) ~ 2*n^2. - Amiram Eldar, Sep 13 2024

A249152 Exponent of 2 in the hyperfactorials: a(n) = A007814(A002109(n)).

Original entry on oeis.org

0, 0, 2, 2, 10, 10, 16, 16, 40, 40, 50, 50, 74, 74, 88, 88, 152, 152, 170, 170, 210, 210, 232, 232, 304, 304, 330, 330, 386, 386, 416, 416, 576, 576, 610, 610, 682, 682, 720, 720, 840, 840, 882, 882, 970, 970, 1016, 1016, 1208, 1208, 1258, 1258, 1362, 1362, 1416, 1416, 1584, 1584, 1642, 1642

Offset: 0

Antti Karttunen, Oct 25 2014

nonn

This is the function ord_2(D*_n) listed in the leftmost column of Table 7.1 in Lagarias & Mehta 2014 paper (on page 19).

Amiram Eldar, Table of n, a(n) for n = 0..10000
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; arXiv preprint, arXiv:1409.4145 [math.NT], 2014-2015.
Luca Onnis, On the p-adic valuation of a hyperfactorial, arXiv:2109.05616 [math.NT], 2021.

Bisection: A249153.
Cf. A174605, A187059, A002109, A007814, A091512, A143157.
Cf. A133457 (binary exponents).

[0] cat [&+[i*Valuation(i, 2):i in [1..n]]:n in [1..60]]; // Marius A. Burtea, Oct 18 2019

with(padic): seq(add(i*ordp(i, 2), i=1..n), n=0..60); # Ridouane Oudra, Oct 17 2019

Table[i=0;Hyperfactorial@n//.x_/;EvenQ@x:>(i++;x/2);i,{n,0,60}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

a(n) = sum(i=1, n, i*valuation(i, 2)); \\ Michel Marcus, Sep 14 2021

a(n) = my(v=binary(n),t=0); forstep(j=#v,1,-1, if(v[j],v[j]=t--,t++)); (n^2 + fromdigits(v,2))>>1; \\ Kevin Ryde, Nov 03 2021

def A249152(n): return sum(i*(~i&i-1).bit_length() for i in range(2,n+1,2)) # Chai Wah Wu, Jul 11 2022

a(n) = 2 * A143157(floor(n/2)).
a(n) = A174605(n) + A187059(n). [Lagarias and Mehta theorem 4.1 for p=2]
a(n) = Sum_{i=1..n} i*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Oct 17 2019
a(n) ~ (n^2+2n)/2 as n -> infinity. - Luca Onnis, Oct 17 2021
a(n) ~ ((A011371(n))^2)/2 as n -> infinity. - Luca Onnis, Nov 02 2021
From Kevin Ryde, Nov 03 2021: (Start)
a(2n) = a(2n+1) = 2*a(n) + n*(n+1).
a(n) = ( n^2 + Sum_{j=1..k} (e[j]-2*j+1) * 2^e[j] )/2, where binary expansion n = 2^e[1] + ... + 2^e[k] with ascending exponents e[1] < e[2] < ... < e[k] (A133457).
(End)
a(n) = Sum_{j=1..floor(log_2(n))} j*2^j*round(n/2^(j+1))^2, for n>=1. - Ridouane Oudra, Oct 01 2022

A143157 Partial sums of A091512.

Original entry on oeis.org

0, 1, 5, 8, 20, 25, 37, 44, 76, 85, 105, 116, 152, 165, 193, 208, 288, 305, 341, 360, 420, 441, 485, 508, 604, 629, 681, 708, 792, 821, 881, 912, 1104, 1137, 1205, 1240, 1348, 1385, 1461, 1500, 1660, 1701, 1785, 1828, 1960, 2005, 2097, 2144, 2384, 2433, 2533, 2584, 2740, 2793, 2901, 2956, 3180

Offset: 0

Gary W. Adamson, Jul 27 2008

			a(4) = 20 = sum of row 4 terms of triangle A143156, (7 + 6 + 4 + 3).
a(4) = 20 = partial sums of first 4 terms of A091512: (1 + 4 + 3 + 12).
a(4) = 20 = Sum_{j=1..4} j*A001511(j) = 1*1 + 2*2 + 3*1 + 4*3.

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A001511, A007814, A091512, A143156, A249152, A249153.

{0}~Join~Accumulate@ Array[IntegerExponent[(2 #)^#, 2] &, 56] (* Michael De Vlieger, Sep 29 2019 *)

def A143157(n): return sum(i*(~i&i-1).bit_length() for i in range(2,2*n+1,2))>>1 # Chai Wah Wu, Jul 11 2022

Partial sums of A091512 = Sum_{j>=1} j*A001511(j), where A001511 is the ruler sequence.
Row sums of triangle A143156.
a(n) = A249152(2*n)/2 = A249153(n) / 2. - Antti Karttunen, Oct 25 2014
a(n) = (1/2)*n*(n + 1) + Sum_{i=1..n} i*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Sep 03 2019; Jan 22 2021
G.f. A(x) satisfies: A(x) = 2*A(x^2)*(1 + x) + x/(1 - x)^3. - Ilya Gutkovskiy, Oct 30 2019
a(n) ~ n^2. - Amiram Eldar, Sep 10 2024

a(0) = 0 prepended and more terms computed by Antti Karttunen, Oct 25 2014

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A069895 2^a(n) divides (2n)^(2n): exponent of 2 in (2n)^(2n).

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A249152 Exponent of 2 in the hyperfactorials: a(n) = A007814(A002109(n)).

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A143157 Partial sums of A091512.

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