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A067745
Numerator of ((3*n - 2)/(n^(2*n - 1)*(2*n - 1)*4^(n - 1))).
11
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
1,3
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
D. S. Mitrinovic and Slavko Simic, Special Functions from Long Ago: 5626, Amer. Math. Monthly 109, (2002), p. 83-84.
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
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 *)
PROG
(PARI) vector(80, n, numerator(((3*n - 2)/(n^(2*n - 1)*(2*n - 1)*4^(n - 1))))) \\ Michel Marcus, Feb 16 2015
(Magma) [Numerator(((3*n - 2)/(n^(2*n - 1)*(2*n - 1)*4^(n - 1)))): n in [1..80]]; // Vincenzo Librandi, Feb 16 2015
CROSSREFS
KEYWORD
easy,frac,nonn
AUTHOR
Marc LeBrun, Jan 29 2002
STATUS
approved