cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A369797 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+2))))).

Original entry on oeis.org

7, 5, 13, 2, 19, 11, 5, 1, 31, 17, 37, 1, 43, 23, 1, 1, 1, 29, 61, 1, 67, 1, 73, 1, 79, 41, 1, 1, 1, 47, 97, 1, 103, 53, 109, 1, 1, 59, 1, 1, 127, 1, 1, 1, 139, 71, 1, 1, 151, 1, 157, 1, 163, 83, 1, 1, 1, 89, 181, 1, 1, 1, 193, 1, 199, 101, 1, 1, 211
Offset: 3

Views

Author

Mohammed Bouras, Feb 25 2024

Keywords

Comments

Conjecture: The sequence contains only 1's and the primes.
Conjecture: The sequence of record values is A002476. - Bill McEachen, Mar 24 2024
a(n) = 1 positions appear to correspond to A334919(m) - 1, m > 2. - Bill McEachen, Aug 05 2024

Examples

			For n=3, 1/(2 - 3/(3 + 2)) = 5/7, so a(3)=7.
For n=4, 1/(2 - 3/(3 - 4/(4 + 2))) = 7/5, so a(4)=5.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 2)))) = 41/13, so a(5)=13.

Links

Crossrefs

Cf. A051403, A356360.

Programs

  • Python
    from math import gcd, factorial
def A369797(n): return (a:=3*n-2)//gcd(a,a*sum(factorial(k) for k in range(n-2))+n*factorial(n-2)>>1) # Chai Wah Wu, Feb 26 2024

Formula

a(n) = (3n - 2)/gcd(3n - 2, A051403(n-2) + 2*A051403(n-3)).

A370726 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+3))))).

Original entry on oeis.org

3, 13, 17, 7, 5, 29, 11, 37, 41, 1, 7, 53, 19, 61, 1, 23, 73, 1, 1, 1, 89, 31, 97, 101, 1, 109, 113, 1, 1, 1, 43, 1, 137, 47, 1, 149, 1, 157, 1, 1, 1, 173, 59, 181, 1, 1, 193, 197, 67, 1, 1, 71, 1, 1, 1, 229, 233, 79, 241, 1, 83, 1, 257, 1, 1, 269, 1, 277
Offset: 3

Views

Author

Mohammed Bouras, Feb 28 2024

Keywords

Comments

Conjecture: The sequence contains only 1's and the primes.
Conjecture: Record values correspond to A002144 (n>3). - Bill McEachen, May 21 2024

Examples

			For n=3, 1/(2 - 3/(3 + 3)) = 2/3, so a(3)=3.
For n=4, 1/(2 - 3/(3 - 4/(4 + 3))) = 17/13, so a(4)=13.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 3)))) = 49/17, so a(5)=17.

Links

Crossrefs

Cf. A051403, A356360, A369797.

Formula

a(n) = (4n - 3)/gcd(4n - 3, A051403(n-2) + 3*A051403(n-3)).

A372761 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+4))))).

Original entry on oeis.org

11, 4, 7, 13, 31, 1, 41, 23, 17, 1, 61, 1, 71, 19, 1, 43, 1, 1, 101, 53, 37, 29, 1, 1, 131, 1, 47, 73, 151, 1, 1, 83, 1, 1, 181, 1, 191, 1, 67, 103, 211, 1, 1, 113, 1, 59, 241, 1, 251, 1, 1, 1, 271, 1, 281, 1, 97, 1, 1, 1, 311, 79, 107, 163, 331, 1, 1, 173, 1
Offset: 3

Views

Author

Mohammed Bouras, May 12 2024

Keywords

Comments

Conjecture 1: Except for 4, the sequence contains only 1's and the primes.
Conjecture 2: Except for 3 and 5, all odd primes appear in the sequence once.
Conjecture: Record values correspond to A030430 (except a(6) = 13). - Bill McEachen, Aug 03 2024

Examples

			For n=3, 1/(2 - 3/(3 + 4)) = 7/11, so a(3)=11.
For n=4, 1/(2 - 3/(3 - 4/(4 + 4))) = 5/4, so a(4)=4.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 4)))) = 19/7, so a(5)=7.
For n=6, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6/(6 + 4))))) = 101/13, so a(6)=13.

Links

Crossrefs

Cf. A051403, A356360, A369797. A370726.

Formula

a(n) = (5n - 4)/gcd(5n - 4, A051403(n-2) + 4*A051403(n-3)).

A372763 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+5))))).

Original entry on oeis.org

13, 19, 5, 31, 37, 43, 7, 11, 61, 67, 73, 79, 17, 1, 97, 103, 109, 23, 11, 127, 1, 139, 29, 151, 157, 163, 1, 1, 181, 1, 193, 199, 41, 211, 1, 223, 229, 47, 241, 1, 1, 1, 53, 271, 277, 283, 1, 59, 1, 307, 313, 1, 1, 331, 337, 1, 349, 71, 1, 367, 373, 379, 1, 1, 397, 1, 409, 83, 421
Offset: 3

Views

Author

Mohammed Bouras, May 12 2024

Keywords

Comments

Conjecture 1: The sequence contains only 1's and the primes.
Conjecture 2: Except for 2 and 3, all primes appear in the sequence once.
Conjecture: Record values correspond to A045375(m), m > 2. - Bill McEachen, Aug 03 2024

Examples

			For n=3, 1/(2 - 3/(3 + 5)) = 8/13, so a(3)=13.
For n=4, 1/(2 - 3/(3 - 4/(4 + 5))) = 23/19, so a(4)=19.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 5)))) = 13/5, so a(5)=5.
For n=6, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6/(6 + 5))))) = 227/31, so a(6)=31.

Links

Crossrefs

Cf. A051403, A356360, A369797. A370726.

Formula

a(n) = (6n - 5)/gcd(6n - 5, A051403(n-2) + 5*A051403(n-3)).
Showing 1-4 of 4 results.

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