A022461 -id:A022461 - cp's OEIS frontend

A333599 a(n) = Fibonacci(n) * Fibonacci(n+1) mod Fibonacci(n+2).

0, 1, 2, 1, 7, 1, 20, 1, 54, 1, 143, 1, 376, 1, 986, 1, 2583, 1, 6764, 1, 17710, 1, 46367, 1, 121392, 1, 317810, 1, 832039, 1, 2178308, 1, 5702886, 1, 14930351, 1, 39088168, 1, 102334154, 1, 267914295, 1, 701408732, 1, 1836311902, 1, 4807526975, 1, 12586269024

Offset: 0

Adnan Baysal, Mar 28 2020

			a(0) = 0*1 mod 1 = 0;
a(1) = 1*1 mod 2 = 1;
a(2) = 1*2 mod 3 = 2;
a(3) = 2*3 mod 5 = 1;
a(4) = 3*5 mod 8 = 7.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, d'Ocagne's Identity.
Index entries for linear recurrences with constant coefficients, signature (-1,3,3,-1,-1).

Equals A035508 interleaved with A000012.

Cf. A182126, A000045, A022461, A072565, A022462.

With[{f = Fibonacci}, Table[Mod[f[n] * f[n+1], f[n+2]], {n, 0, 50}]] (* Amiram Eldar, Mar 28 2020 *)

a(n) = if (n % 2, 1, fibonacci(n+2) - 1); \\ Michel Marcus, Mar 29 2020

concat(0, Vec(x*(1 + 3*x - x^3) / ((1 + x)*(1 + x - x^2)*(1 - x - x^2)) + O(x^45))) \\ Colin Barker, Mar 29 2020

def a(n):
    f1 = 0
    f2 = 1
    for i in range(n):
        f = f1 + f2
        f1 = f2
        f2 = f
    return (f1 * f2) % (f1 + f2)

a(2n+1) = 1, and a(2n) = F(2n+2) - 1, and lim(a(2n+2)/a(2n)) = phi^2 by d'Ocagne's identity.
a(n) = F(n) * F(n+1) mod (F(n) + F(n+1)) since F(n+2) := F(n+1) + F(n).
From Colin Barker, Mar 28 2020: (Start)
G.f.: x*(1 + 3*x - x^3) / ((1 + x)*(1 + x - x^2)*(1 - x - x^2)).
a(n) = -a(n-1) + 3*a(n-2) + 3*a(n-3) - a(n-4) - a(n-5) for n>4.
(End)

A072565 a(n) = prime(n+1)*prime(n+2)+1 mod prime(n), where prime(n) is the n-th prime.

Original entry on oeis.org

0, 0, 3, 4, 2, 12, 13, 3, 3, 17, 30, 25, 13, 41, 26, 49, 17, 0, 25, 17, 61, 41, 2, 8, 25, 13, 25, 13, 73, 27, 41, 49, 25, 121, 17, 73, 61, 41, 73, 49, 25, 121, 13, 25, 29, 90, 193, 25, 13, 41, 49, 25, 161, 73, 73, 49, 17, 61, 25, 25, 241, 253, 25, 13, 73, 281, 97, 121, 13

Offset: 1

G. L. Honaker, Jr., Aug 06 2002

nonn

			a(18) = prime(19)*prime(20)+1 mod prime(18) = 67*71+1 mod 61 = 0.

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer-Verlag, NY, (2002 printing), Research problem 1.85, p. 73.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040, A023523, A022461.

P:=Filtered([1..1000],IsPrime);;
List([1..70],n->(P[n+1]*P[n+2]+1) mod P[n]); # Muniru A Asiru, Mar 09 2018

[(NthPrime(n+1)*NthPrime(n+2)+1) mod NthPrime(n): n in [1..100]]; // Vincenzo Librandi, Feb 28 2018

p:=ithprime; seq((p(n+1)*p(n+2)+1) mod p(n),n=1..70); # Muniru A Asiru, Mar 09 2018

a[n_] := Mod[Prime[n+1] Prime[n+2] + 1, Prime[n]]
Mod[#[[2]]#[[3]]+1,#[[1]]]&/@Partition[Prime[Range[80]],3,1] (* Harvey P. Dale, Dec 19 2018 *)

a(n) = (prime(n+1)*prime(n+2) + 1) % prime(n); \\ Michel Marcus, Feb 28 2018

a(n) = A023523(n+1) mod A000040(n). - Michel Marcus, Feb 28 2018

Edited by Dean Hickerson and Robert G. Wilson v, Aug 10 2002

A182126 a(n) = prime(n)*prime(n+1) mod prime(n+2).

Original entry on oeis.org

1, 1, 2, 12, 7, 12, 1, 2, 16, 11, 40, 12, 24, 7, 13, 16, 48, 40, 12, 48, 40, 60, 15, 48, 12, 24, 12, 24, 125, 72, 60, 16, 120, 24, 48, 72, 40, 60, 72, 16, 120, 24, 24, 12, 168, 65, 64, 12, 24, 60, 16, 120, 96, 72, 72, 16, 48, 40, 12, 120, 29, 72, 12, 24, 252

Offset: 1

Alex Ratushnyak, Apr 13 2012

Conjecture: for x>10^9, the most frequent value in a(n), n=0...x, has form 120*k.
Let b = prime(n+2) - prime(n) and c = prime(n+2) - prime(n+1). Conjecture: for n > 61, a(n) = b*c. This holds up to n = 9 * 10^16. - Charles R Greathouse IV, May 11 2012
With b and c as above, a(n) = b*c if and only if b*c < prime(n+2). Cramér's conjecture implies this is true for all sufficiently large n. - Robert Israel, Jun 19 2017

			(2*3) mod 5 = 1, (3*5) mod 7 = 1, (5*7) mod 11 = 2, (7*11) mod 13 = 12.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A022461, A022462.

a182126 n = a182126_list !! (n-1)
a182126_list = zipWith3 (\p p' p'' -> mod (p * p') p'')
                  a000040_list (tail a000040_list) (drop 2 a000040_list)
-- Reinhard Zumkeller, Apr 23 2012

[NthPrime(n)*NthPrime(n+1) mod NthPrime(n+2): n in [1..70]]; // Vincenzo Librandi, Jun 20 2017

P:= [seq(ithprime(i),i=1..102)]:
seq(P[i]*P[i+1] mod P[i+2], i=1..100); # Robert Israel, Jun 19 2017

Mod[#[[1]]#[[2]],#[[3]]]&/@Partition[Prime[Range[70]],3,1] (* Harvey P. Dale, Sep 30 2015 *)

p=2;q=3;forprime(r=5,1e3,print1(p*q%r", ");p=q;q=r) \\ Charles R Greathouse IV, May 11 2012

cp's OEIS Frontend

A333599 a(n) = Fibonacci(n) * Fibonacci(n+1) mod Fibonacci(n+2).

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A072565 a(n) = prime(n+1)*prime(n+2)+1 mod prime(n), where prime(n) is the n-th prime.

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A182126 a(n) = prime(n)*prime(n+1) mod prime(n+2).

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