A028877 -id:A028877 - cp's OEIS frontend

A028875 a(n) = n^2 - 5.

-5, -4, -1, 4, 11, 20, 31, 44, 59, 76, 95, 116, 139, 164, 191, 220, 251, 284, 319, 356, 395, 436, 479, 524, 571, 620, 671, 724, 779, 836, 895, 956, 1019, 1084, 1151, 1220, 1291, 1364, 1439, 1516, 1595, 1676, 1759, 1844, 1931, 2020, 2111, 2204, 2299, 2396

Offset: 0

Patrick De Geest, Dec 11 1999

a(n) gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 20 for b = 2*n. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013

For n>2, a(n) represents the area of the triangle created by the three points defined with coordinates: (n-3,n-2), ((n-1)*n/2,n*(n+1)/2), and ((n+1)^2, (n+2)^2). - J. M. Bergot, May 22 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Near-Square Prime.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028877 (subset of primes).

[n^2-5: n in [0..50]]; // Wesley Ivan Hurt, May 22 2014

A028875:=n->n^2-5; seq(A028875(n), n=0..100); # Wesley Ivan Hurt, Nov 13 2013

Range[0, 49]^2 - 5 (* Alonso del Arte, Aug 27 2013 *)

a(n)=n^2-5 \\ Charles R Greathouse IV, Oct 07 2015

From R. J. Mathar, Apr 28 2008: (Start)
G.f.: x^3*(4 - x - x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = 2*n + a(n-1) + 5, with n>0, a(0)=4. - Vincenzo Librandi, Aug 05 2010
a(-n) = a(n). - Michael Somos, May 26 2014
E.g.f.: (x^2 + x - 5)*exp(x). - G. C. Greubel, Aug 19 2017
From Amiram Eldar, Nov 04 2020: (Start)
Sum_{n>=0} 1/a(n) = -(1 + sqrt(5)*Pi*cot(sqrt(5)*Pi))/10.
Sum_{n>=0} (-1)^n/a(n) = -(1 + sqrt(5)*Pi*cosec(sqrt(5)*Pi))/10. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = sqrt(6/5)*sin(sqrt(6)*Pi)/sin(sqrt(5)*Pi).
Product_{n>=3} (1 + 1/a(n)) = sqrt(5)*Pi/(6*sin(sqrt(5)*Pi)). (End)

A028876 Numbers k such that k^2 - 5 is prime.

Original entry on oeis.org

4, 6, 8, 12, 14, 16, 22, 24, 32, 34, 36, 38, 42, 44, 46, 52, 58, 64, 72, 74, 78, 82, 94, 102, 112, 116, 122, 132, 144, 152, 164, 166, 168, 174, 176, 182, 184, 186, 188, 198, 204, 212, 222, 226, 232, 234, 236, 252, 262, 264, 278, 284, 288, 292, 298, 302, 318, 324

Offset: 1

Patrick De Geest

G. C. Greubel, Table of n, a(n) for n = 1..1000
P. De Geest, Palindromic Quasipronics of the form n(n+x)
Eric Weisstein's World of Mathematics, Near-Square Prime

Cf. A028875, A028877.

[n: n in [2..1000] |IsPrime(n^2 - 5 )]; // Vincenzo Librandi, Nov 18 2010

Select[Range[400], PrimeQ[#^2 - 5] &] (* Alonso del Arte, Aug 27 2013 *)

is(n)=isprime(n^2-5) \\ Charles R Greathouse IV, Jun 06 2017

A352804 a(n) = A028876(n)/2; numbers k such that 4*k^2 - 5 is prime.

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 11, 12, 16, 17, 18, 19, 21, 22, 23, 26, 29, 32, 36, 37, 39, 41, 47, 51, 56, 58, 61, 66, 72, 76, 82, 83, 84, 87, 88, 91, 92, 93, 94, 99, 102, 106, 111, 113, 116, 117, 118, 126, 131, 132, 139, 142, 144, 146, 149, 151, 159, 162, 171, 177, 179

Offset: 1

Jianing Song, Apr 04 2022

A028876 with common factor 2 removed.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A028876, A028877 (the resulting primes), A352805.

Mathematica
```
Select[Range[200], PrimeQ[4*#^2 - 5] &]
```
PARI
```
isA352804(n)=isprime(4*n^2-5)
```

A363347 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(-4))))).

Original entry on oeis.org

11, 5, 31, 11, 59, 19, 19, 29, 139, 41, 191, 1, 251, 71, 29, 89, 79, 109, 479, 131, 571, 31, 61, 181, 41, 1, 179, 239, 1019, 271, 1151, 61, 1291, 1, 1439, 379, 1, 419, 1759, 461, 1931, 101, 2111, 1, 1, 599, 499, 59, 2699, 701, 71, 151, 101, 811

Offset: 3

Mohammed Bouras, May 28 2023

nonn

Conjecture 1: Every term of this sequence is either a prime or 1.
Conjecture 2: The sequence contains all prime numbers which end with a 1 or 9.
Conjecture 3: Except for 5, the primes all appear exactly twice.
Conjecture: The sequence of record values is A028877. - Bill McEachen, May 20 2024

			For n=3, 1/(2 - 3/(-4)) = 4/11, so a(3) = 11.
For n=4, 1/(2 - 3/(3 - 4/(-4))) = 4/5, so a(4) = 5.
For n=5, 1/(2 - 3/(3 - 4/(4 -5/(-4)))) = 47/31, so a(5) = 31.
a(3) = a(6) = 3 + 6 + 2 = 11.
a(5) = a(24) = 5 + 24 + 2 = 31.
a(7) = a(50) = 7 + 50 + 2 = 59.

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

Cf. A006530, A051403, A229525, A356247, A028877.

a(n) = (n^2 + 2*n - 4)/gcd(n^2 + 2*n - 4, 4*A051403(n-3) + n*A051403(n-4)).
a(n) = gpf(n^2 + 2*n - 4) if gpf(n^2 + 2*n - 4) > n, otherwise a(n) = 1 (where gpf(n) denotes the greatest prime factor of n).
If n != m and a(n) = a(m) != 1, then we have:
a(n) = n + m + 2.
a(n) = gcd(n^2 + 2*n - 4, m^2 + 2*m - 4).

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A028875 a(n) = n^2 - 5.

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A028876 Numbers k such that k^2 - 5 is prime.

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A352804 a(n) = A028876(n)/2; numbers k such that 4*k^2 - 5 is prime.

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A363347 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(-4))))).

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