A051417 -id:A051417 - cp's OEIS frontend

A025547 Least common multiple of {1,3,5,...,2n-1}.

1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675

Offset: 1

Clark Kimberling

This sequence coincides with the sequence f(n) = denominator of 1 + 1/3 + 1/5 + 1/7 + ... + 1/(2n-1) iff n <= 38. But a(39) = 6414924694381721303722858446525, f(39) = 583174972216520118520259858775. - T. D. Noe, Aug 04 2004 [See A350670(n-1).]
Coincides for n=1..42 with the denominators of a series for Pi*sqrt(2)/4 and then starts to differ. See A127676.
a(floor((n+1)/2)) = gcd(a(n), A051426(n)). - Reinhard Zumkeller, Apr 25 2011
A051417(n) = a(n+1)/a(n).

T. D. Noe, Table of n, a(n) for n = 1..200
Yue-Wu Li and Feng Qi, A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
Eric Weisstein's World of Mathematics, Jeep Problem, Pi, Pi Continued Fraction, Least Common Multiple
Wikipedia, Least common multiple
Index entries for sequences related to lcm's

Cf. A007509, A025550, A075135. The numerators are in A074599.
Cf. A003418 (LCM of {1..n}).
Cf. A005408, A350670.

a025547 n = a025547_list !! (n-1)
a025547_list = scanl1 lcm a005408_list
-- Reinhard Zumkeller, Oct 25 2013, Apr 25 2011

A025547:=proc(n) local i,t1; t1:=1; for i from 1 to n do t1:=lcm(t1,2*i-1); od: t1; end;
f := n->denom(add(1/(2*k-1),k=0..n)); # a different sequence!

a = 1; Join[{1}, Table[a = LCM[a, n], {n, 3, 125, 2}]] (* Zak Seidov, Jan 18 2011 *)
nn=30;With[{c=Range[1,2*nn,2]},Table[LCM@@Take[c,n],{n,nn}]] (* Harvey P. Dale, Jan 27 2013 *)

a(n)=lcm(vector(n,k,2*k-1)) \\ Charles R Greathouse IV, Nov 20 2012

# generates initial segment of sequence
from math import gcd
from itertools import accumulate
def lcm(a, b): return a * b // gcd(a, b)
def aupton(nn): return list(accumulate((2*i+1 for i in range(nn)), lcm))
print(aupton(23)) # Michael S. Branicky, Mar 28 2022

A088961 Zigzag matrices listed entry by entry.

Original entry on oeis.org

3, 5, 5, 5, 10, 14, 14, 7, 14, 21, 21, 7, 21, 35, 42, 48, 27, 9, 48, 69, 57, 36, 27, 57, 78, 84, 9, 36, 84, 126, 132, 165, 110, 44, 11, 165, 242, 209, 121, 55, 110, 209, 253, 220, 165, 44, 121, 220, 297, 330, 11, 55, 165, 330, 462

Offset: 1

Paul Boddington, Oct 28 2003

For each n >= 1 the n X n matrix Z(n) is constructed as follows. The i-th row of Z(n) is obtained by generating a hexagonal array of numbers with 2*n+1 rows, 2*n numbers in the odd numbered rows and 2*n+1 numbers in the even numbered rows. The first row is all 0's except for two 1's in the i-th and the (2*n+1-i)th positions. The remaining rows are generated using the same rule for generating Pascal's triangle. The i-th row of Z(n) then consists of the first n numbers in the bottom row of our array.
For example the top row of Z(2) is [5,5], found from the array:
. 1 0 0 1
1 1 0 1 1
. 2 1 1 2
2 3 2 3 2
. 5 5 5 5
Zigzag matrices have remarkable properties. Here is a selection:
1) Z(n) is symmetric.
2) det(Z(n)) = A085527(n).
3) tr(Z(n)) = A033876(n-1).
4) If 2*n+1 is a power of a prime p then all entries of Z(n) are multiples of p.
5) If 4*n+1 is a power of a prime p then the dot product of any two distinct rows of Z(n) is a multiple of p.
6) It is always possible to move from the bottom left entry of Z(n) to the top right entry using only rightward and upward moves and visiting only odd numbers.
A001700(n) = last term of last row of Z(n): a(A000330(n-1)) = A001700(n); A230585(n) = first term of first row of Z(n): a(A056520(n-1)) = A230585(n); A051417(n) = greatest common divisor of entries of Z(n). - Reinhard Zumkeller, Oct 25 2013

			The first five values are 3, 5, 5, 5, 10 because the first two zigzag matrices are [[3]] and [[5,5],[5,10]].

Reinhard Zumkeller, Matrices Z(n): n = 1..30, flattened

Cf. A085527, A003876.

a088961 n = a088961_list !! (n-1)
a088961_list = concat $ concat $ map f [1..] where
   f x = take x $ g (take x (1 : [0,0..])) where
     g us = (take x $ g' us) : g (0 : init us)
     g' vs = last $ take (2 * x + 1) $
                    map snd $ iterate h (0, vs ++ reverse vs)
   h (p,ws) = (1 - p, drop p $ zipWith (+) ([0] ++ ws) (ws ++ [0]))
-- Reinhard Zumkeller, Oct 25 2013

Flatten[Table[Binomial[2n,n+j-i]-Binomial[2n,n+i+j]+ Binomial[2n, 3n+1-i-j], {n,5},{i,n},{j,n}]] (* Harvey P. Dale, Dec 15 2011 *)

The ij entry of Z(n) is binomial(2*n, n+j-i) - binomial(2*n, n+i+j) + binomial(2*n, 3*n+1-i-j).

A356360 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).

Original entry on oeis.org

5, 7, 3, 11, 13, 1, 17, 19, 1, 23, 1, 1, 29, 31, 1, 1, 37, 1, 41, 43, 1, 47, 1, 1, 53, 1, 1, 59, 61, 1, 1, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 1, 1, 1, 127, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1, 149, 151, 1, 1, 157, 1, 1, 163, 1, 167

Offset: 3

Mohammed Bouras, Oct 15 2022

nonn

Conjecture: The sequence contains only 1's and the primes.
Similar continued fraction to A356247.
Same as A128059(n), A145737(n-1) and A097302(n-2) for n > 5.
a(n) = 1 positions appear to correspond to A104275(m), m > 2. Conjecture: all odd primes are seen in order after 11. - Bill McEachen, Aug 05 2024

Mohammed Bouras, The Distribution Of Prime Numbers And The Continued Fractions, (2022).

Cf. A051403, A051417, A097302, A128059, A145737, A356247.

Cf. A104275.

from math import gcd, factorial
def A356360(n): return (a:=(n<<1)-1)//gcd(a, a*sum(factorial(k) for k in range(n-2))+n*factorial(n-2)>>1) # Chai Wah Wu, Feb 26 2024

For n >= 3, the formula of the continued fraction is as follows:
(A051403(n-2) + A051403(n-3))/(2n - 1) = 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).
a(n) = (2n - 1)/gcd(2n - 1, A051403(n-2) + A051403(n-3)).
From the conjecture: Except for n = 5, a(n)= 2n - 1 if 2n-1 is prime, 1 otherwise.

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A025547 Least common multiple of {1,3,5,...,2n-1}.

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A088961 Zigzag matrices listed entry by entry.

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A356360 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).

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