A143157 -id:A143157 - cp's OEIS frontend

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

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

A249153 Exponent of 2 in the hyperfactorial of 2n: a(n) = A007814(A002109(2n)).

Original entry on oeis.org

0, 2, 10, 16, 40, 50, 74, 88, 152, 170, 210, 232, 304, 330, 386, 416, 576, 610, 682, 720, 840, 882, 970, 1016, 1208, 1258, 1362, 1416, 1584, 1642, 1762, 1824, 2208, 2274, 2410, 2480, 2696, 2770, 2922, 3000, 3320, 3402, 3570, 3656, 3920, 4010, 4194, 4288, 4768, 4866, 5066, 5168, 5480, 5586, 5802, 5912

Offset: 0

Antti Karttunen, Oct 25 2014

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Bisection of A249152.

Cf. A002109, A007814, A143157, A069895 (first differences).

Table[IntegerExponent[Hyperfactorial[2*n], 2], {n, 0, 55}] (* Amiram Eldar, Sep 10 2024 *)

from sympy import multiplicity
A249153_list, n = [0], 0
for i in range(2,20002,2):
    n += multiplicity(2,i)*i
    A249153_list.append(n) # Chai Wah Wu, Aug 21 2015

a(n) = A249152(2*n) = A007814(A002109(2*n)).
a(n) = 2*A143157(n).
a(n) ~ 2*n^2. - Amiram Eldar, Sep 10 2024

A143156 Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.

Original entry on oeis.org

1, 3, 2, 4, 3, 1, 7, 6, 4, 3, 8, 7, 5, 4, 1, 10, 9, 7, 6, 3, 2, 11, 10, 8, 7, 4, 3, 1, 15, 14, 12, 11, 8, 7, 5, 4, 16, 15, 13, 12, 9, 8, 6, 5, 1, 18, 17, 15, 14, 11, 10, 8, 7, 3, 2, 19, 18, 16, 15, 12, 11, 9, 8, 4, 3, 1, 22, 21, 19, 18, 15, 14, 12, 11, 7, 6, 4, 3

Offset: 1

Gary W. Adamson, Jul 27 2008

Row sums give A143157.
Left border gives A005187.
Right border gives A001511.

			First few rows of the triangle =
        k=1 k=2 k=3 k=4 k=5 k=6 k=7
  n=1:   1;
  n=2:   3,  2;
  n=3:   4,  3,  1;
  n=4:   7,  6,  4,  3;
  n=5:   8,  7,  5,  4,  1;
  n=6:  10,  9,  7,  6,  3,  2;
  n=7:  11, 10,  8,  7,  4,  3,  1;
  ...
Row 6 = (10, 9, 7, 6, 3, 2) = partial sums of the first 6 terms of the ruler sequence, starting from the right: (1, 2, 1, 3, 1, 2,...).

Cf. A001511, A000012, A091512, A005187.

T(n,k) = k--; 2*(n-k) - hammingweight(n) + hammingweight(k); \\ Kevin Ryde, Oct 07 2021

Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.
From Kevin Ryde, Oct 07 2021: (Start)
T(n,k) = A005187(n) - A005187(k-1).
G.f.: (V(x) - V(x*y)) * y/((1-x)*(1-y)) where V(x) is the g.f. of A001511.
(End)

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

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A249153 Exponent of 2 in the hyperfactorial of 2n: a(n) = A007814(A002109(2n)).

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Author

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Programs

Mathematica

Python

Formula

A143156 Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.

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