A067745 Numerator of ((3*n - 2)/(n^(2*n - 1)*(2*n - 1)*4^(n - 1))).

1, 1, 7, 5, 13, 1, 19, 11, 25, 7, 31, 17, 37, 5, 43, 23, 49, 13, 55, 29, 61, 1, 67, 35, 73, 19, 79, 41, 85, 11, 91, 47, 97, 25, 103, 53, 109, 7, 115, 59, 121, 31, 127, 65, 133, 17, 139, 71, 145, 37, 151, 77, 157, 5, 163, 83, 169, 43, 175, 89, 181, 23, 187, 95, 193, 49, 199

Offset: 1

Author

Marc LeBrun, Jan 29 2002

Keywords

• easy
• frac
• nonn

Comments

Conjecture: Odd part of 3n-2. - Ralf Stephan, Nov 18 2010

Conjecture is true. Note that gcd(3n-2,2n-1)=1 (because 2(3n-2)-3(2n-1) = -1) and gcd(3n-2,n) = 1 or 2. If 2^k | (3n-2), then k <= log_2(3n-2) < (n-1)/2 for n >= 11. So only the cases n <= 10 need to be checked individually. - Robert Israel, May 16 2017

This sequence is equivalent to A165355 where each element is reduced by the highest possible power of two. - Joe Slater, Nov 30 2016

Selecting each odd term gives b(n) = 6n+1 (A016921). A075677 is the even bisection of this sequence, while this sequence is the odd bisection of A075677. - Cory Kalm, Apr 29 2021

Numerator of n/2^n + (n-1)/2^(n-1), two Oresme numbers. - Paul Curtz, Dec 07 2021

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• D. S. Mitrinovic and Slavko Simic, Special Functions from Long Ago: 5626, Amer. Math. Monthly 109, (2002), p. 83-84.

Crossrefs

Cf. A067746, A165355, A000265, A075677, A016921, A007310 (range).

Cf. A075101.

Programs

Magma

[Numerator(((3*n - 2)/(n^(2*n - 1)*(2*n - 1)*4^(n - 1)))): n in [1..80]]; // Vincenzo Librandi, Feb 16 2015

Maple

f:= n -> (3*n-2)/2^padic:-ordp(3*n-2,2):
map(f, [$1..100]); # Robert Israel, May 16 2017

Mathematica

(* Assuming the above conjecture: *)
a067745[n_] := (3*n - 2)/2^IntegerExponent[3*n - 2, 2]; Table[a067745[n], {n, 67}] (* L. Edson Jeffery, Feb 15 2015 *)

PARI

vector(80, n, numerator(((3*n - 2)/(n^(2*n - 1)*(2*n - 1)*4^(n - 1))))) \\ Michel Marcus, Feb 16 2015

Formula

Assuming the above conjecture, a(n) = a((8+(3*n-2)*4^k)/12), for all k >= 1. - L. Edson Jeffery, Feb 15 2015

a(n) = A000265(A165355(n-1)). - Joe Slater, Nov 30 2016

a(n) = A000265(3*n-2). - R. J. Mathar, Aug 23 2020

a(n) = A075677(2*n-1). a(2*n) = A075677(n); a(2*n-1) = A016921(n). - Cory Kalm, May 03 2021

Sum_{k=1..n} a(k) ~ n^2. - Amiram Eldar, Aug 26 2024

G.f.: Sum_{k>=1} ((3 + 2*(-1)^(k + 1))*x^(3*2^(k - 1) - (2*(-2)^(k - 1))/3 - 1/3) + (3 - 2*(-1)^(k + 1))*x^(2^(k - 1)*(3 + 2*(-1)^k)/3 - 1/3))/(x^(2^(k + 1)) - 2*x^(2^k) + 1). - Miles Wilson, Jan 12 2025