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

0,3

Comments

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).

Links

Table of n, a(n) for n=0..59.

Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Luca Onnis, On the p-adic valuation of a hyperfactorial, arXiv:2109.05616 [math.NT], 2021.

Formula

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

Maple

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

Mathematica

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

Prog

(Scheme, two alternative implementations)
(define (A249152 n) (A007814 (A002109 n)))
(define (A249152 n) (* 2 (A143157 (floor->exact (/ n 2)))))
(Magma) [0] cat [&+[i*Valuation(i, 2):i in [1..n]]:n in [1..60]]; // Marius A. Burtea, Oct 18 2019
(PARI) a(n) = sum(i=1, n, i*valuation(i, 2)); \\ Michel Marcus, Sep 14 2021
(PARI) 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
(Python)
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

Crossrefs

Bisection: A249153.

Cf. A174605, A187059, A002109, A007814, A091512, A143157.

Cf. A133457 (binary exponents).

Sequence in context: A019241 A168295 A309751 * A216708 A032005 A147801

Adjacent sequences: A249149 A249150 A249151 * A249153 A249154 A249155

Keyword

nonn

Author

Antti Karttunen, Oct 25 2014

Status

approved