cp's OEIS Frontend

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

%I A249152 #62 Sep 10 2024 03:01:52
%S A249152 0,0,2,2,10,10,16,16,40,40,50,50,74,74,88,88,152,152,170,170,210,210,
%T A249152 232,232,304,304,330,330,386,386,416,416,576,576,610,610,682,682,720,
%U A249152 720,840,840,882,882,970,970,1016,1016,1208,1208,1258,1258,1362,1362,1416,1416,1584,1584,1642,1642
%N A249152 Exponent of 2 in the hyperfactorials: a(n) = A007814(A002109(n)).
%C A249152 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).
%H A249152 Amiram Eldar, <a href="/A249152/b249152.txt">Table of n, a(n) for n = 0..10000</a>
%H A249152 Jeffrey C. Lagarias and Harsh Mehta, <a href="https://doi.org/10.1142/S1793042116500044">Products of binomial coefficients and unreduced Farey fractions</a>, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; <a href="http://arxiv.org/abs/1409.4145">arXiv preprint</a>, arXiv:1409.4145 [math.NT], 2014-2015.
%H A249152 Luca Onnis, <a href="https://arxiv.org/abs/2109.05616">On the p-adic valuation of a hyperfactorial</a>, arXiv:2109.05616 [math.NT], 2021.
%F A249152 a(n) = 2 * A143157(floor(n/2)).
%F A249152 a(n) = A174605(n) + A187059(n). [Lagarias and Mehta theorem 4.1 for p=2]
%F A249152 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
%F A249152 a(n) ~ (n^2+2n)/2 as n -> infinity. - _Luca Onnis_, Oct 17 2021
%F A249152 a(n) ~ ((A011371(n))^2)/2 as n -> infinity. - _Luca Onnis_, Nov 02 2021
%F A249152 From _Kevin Ryde_, Nov 03 2021: (Start)
%F A249152 a(2n) = a(2n+1) = 2*a(n) + n*(n+1).
%F A249152 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).
%F A249152 (End)
%F A249152 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
%p A249152 with(padic): seq(add(i*ordp(i, 2), i=1..n), n=0..60); # _Ridouane Oudra_, Oct 17 2019
%t A249152 Table[i=0;Hyperfactorial@n//.x_/;EvenQ@x:>(i++;x/2);i,{n,0,60}] (* _Giorgos Kalogeropoulos_, Oct 28 2021 *)
%o A249152 (Scheme, two alternative implementations)
%o A249152 (define (A249152 n) (A007814 (A002109 n)))
%o A249152 (define (A249152 n) (* 2 (A143157 (floor->exact (/ n 2)))))
%o A249152 (Magma) [0] cat [&+[i*Valuation(i, 2):i in [1..n]]:n in [1..60]]; // _Marius A. Burtea_, Oct 18 2019
%o A249152 (PARI) a(n) = sum(i=1, n, i*valuation(i, 2)); \\ _Michel Marcus_, Sep 14 2021
%o A249152 (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
%o A249152 (Python)
%o A249152 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
%Y A249152 Bisection: A249153.
%Y A249152 Cf. A174605, A187059, A002109, A007814, A091512, A143157.
%Y A249152 Cf. A133457 (binary exponents).
%K A249152 nonn
%O A249152 0,3
%A A249152 _Antti Karttunen_, Oct 25 2014