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.

A370388 First appearance of prime_i in A369797 or 0 if no such appearance occurs, with p_0 being 1. (A008578 but with a different offset.)

Original entry on oeis.org

10, 6, 0, 4, 3, 8, 5, 12, 7, 16, 20, 11, 13, 28, 15, 32, 36, 40, 21, 23, 48, 25, 27, 56, 60, 33, 68, 35, 72, 37, 76, 43, 88, 92, 47, 100, 51, 53, 55, 112, 116, 120, 61, 128, 65, 132, 67, 71, 75, 152, 77, 156, 160, 81, 168, 172, 176, 180, 91, 93, 188, 95, 196, 103, 208, 105, 212
Offset: 0

Views

Author

Robert G. Wilson v, Feb 28 2024

Keywords

Comments

As noted in A369797, a(n) is either 1 or a prime. But not mentioned is that all primes appear once, with two exceptions. The second prime, 3, never appears and the third prime, 5, appears twice, at 4 and 9.
A denominator of 1 in A369797 appears about 81% of the time.
The graph of this sequence has three broken lines, one at a(n) = 1, the second at ~3n/2 and the third at ~3n.

Crossrefs

Cf. A008578, A369797, A039701, A008864, A283419.

Programs

  • Mathematica
    b[n_] := b[n] = (n + 2) (b[n - 1] - b[n - 2]); b[-1] = 0; b[0] = 1; a[n_] := a[n] = (3 n - 2)/GCD[3 n - 2, b[n - 2] + 2 b[n - 3]]; Flatten[ Table[ FirstPosition[ Table[ a[n], {n, 3, 220}], n], {n, Join[{1}, Prime@ Range@ 67]}]] + 2

Formula

Conjectures from Ridouane Oudra, Apr 26 2025: (Start)
a(n) = (2/3) mod prime(n), for n>2.
a(n) = (prime(n)*A039701(n) + 2)/3, for n>2.
a(n) = A008864(n) - A283419(n), for n>2. (End)

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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