A338794 -id:A338794 - cp's OEIS frontend

A339461 Number of Fibonacci divisors of n^2 + 1.

1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 2, 2, 4, 1, 2, 1, 3, 3, 2, 1, 4, 2, 3, 1, 2, 1, 3, 2, 2, 1, 3, 2, 3, 3, 2, 1, 3, 2, 2, 1, 2, 2, 3, 2, 2, 1, 5, 2, 2, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 2, 3, 2, 2, 4, 1, 2, 1, 3, 2, 2, 1, 3, 2, 4, 1, 2, 2

Offset: 0

Michel Lagneau, Dec 06 2020

nonn

			a(13) = 4 because the divisors of 13^2 + 1 = 170 are {1, 2, 5, 10, 17, 34, 85, 170} with 4 Fibonacci divisors: 1, 2, 5 and 34.

Michel Marcus, Table of n, a(n) for n = 0..10000

Cf. A000045, A002522, A005086, A005478, A005574, A128428, A338762, A338794.

with(numtheory):with(combinat,fibonacci):nn:=100:F:={}:
for k from 1 to nn do:
  F:=F union {fibonacci(k)}:
od:
   for n from 0 to 90 do:
    f:=n^2+1:d:=divisors(f):
    lst:= F intersect d: n1:=nops(lst):printf(`%d, `,n1):
   od:

Array[DivisorSum[#^2 + 1, 1 &, Or @@ Map[IntegerQ@ Sqrt[#] &, 5 #^2 + 4 {-1, 1}] &] &, 105, 0] (* Michael De Vlieger, Dec 07 2020 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || issquare(k-8);
a(n) = sumdiv(n^2+1, d, isfib(d)); \\ Michel Marcus, Dec 06 2020

a(A005574(n)) = 1 for n > 2.

a(n) = A005086(A002522(n)). - Michel Marcus, Dec 06 2020

A335568 a(n) is the number m such that F(m) is the greatest prime Fibonacci divisor of F(n)^2 + 1 where F(n) is the n-th Fibonacci number, or 0 if no such prime factor exists.

Original entry on oeis.org

3, 3, 5, 5, 7, 7, 5, 7, 11, 11, 13, 13, 11, 13, 17, 17, 5, 17, 17, 7, 23, 23, 7, 23, 23, 5, 29, 29, 3, 29, 29, 11, 7, 11, 11, 7, 13, 13, 0, 13, 43, 43, 5, 43, 47, 47, 7, 47, 47, 17, 7, 17, 17, 11, 3, 11, 11, 3, 3, 0, 7, 7, 13, 13, 7, 13, 23, 23, 0, 23, 23, 0, 5

Offset: 1

Chai Wah Wu, Nov 20 2020

nonn

Fibonacci index of the terms in A338762.

All terms are prime or 0. - Alois P. Heinz, Nov 21 2020

			a(10) = 11 because F(10)^2 + 1 = 55^2 + 1 = 3026 = 2*17*89 and 89 = F(11) is the greatest prime Fibonacci divisor of 3026.

Chai Wah Wu, Table of n, a(n) for n = 1..20000

Cf. A000040, A000045, A005478, A245306, A338762, A338794 (indices of the 0's).

a:= proc(n) local i, F, m, t; F, m, t:=
      [1, 2], 0, (<<0|1>, <1|1>>^n)[2, 1]^2+1;
      for i from 3 while F[2]<=t do if isprime(F[2]) and
        irem(t, F[2])=0 then m:=i fi; F:= [F[2], F[1]+F[2]]
      od; m
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 21 2020

a[n_] := Module[{i, F = {1, 2}, m = 0, t}, t = MatrixPower[{{0, 1}, {1, 1}}, n][[2, 1]]^2 + 1; For[i = 3, F[[2]] <= t, i++, If[PrimeQ[F[[2]]] && Mod[t, F[[2]]] == 0, m = i]; F = {F[[2]], F[[1]] + F[[2]]}]; m];
Array[a, 100] (* Jean-François Alcover, Dec 01 2020, after Alois P. Heinz *)

A000045(a(n)) = A338762(n).

A339082 a(n) is the number m such that F(prime(m)) is the greatest prime Fibonacci divisor of F(n)^2 + 1 where F(n) is the n-th Fibonacci number, or 0 if no such prime factor exists.

Original entry on oeis.org

2, 2, 3, 3, 4, 4, 3, 4, 5, 5, 6, 6, 5, 6, 7, 7, 3, 7, 7, 4, 9, 9, 4, 9, 9, 3, 10, 10, 2, 10, 10, 5, 4, 5, 5, 4, 6, 6, 0, 6, 14, 14, 3, 14, 15, 15, 4, 15, 15, 7, 4, 7, 7, 5, 2, 5, 5, 2, 2, 0, 4, 4, 6, 6, 4, 6, 9, 9, 0, 9, 9, 0, 3, 3, 5, 5, 3, 5, 5, 2, 23, 23, 7

Offset: 1

Chai Wah Wu, Nov 24 2020

nonn

If a(n) > 0, then prime(a(n)) = A335568(n).

			a(15) = 7 because F(15)^2 + 1 = 610^2 + 1 = 372101 = 233*1597, 1597 = F(17) is the greatest prime Fibonacci divisor of 372101 and 17 is the 7th prime.

Chai Wah Wu, Table of n, a(n) for n = 1..30000

Cf. A000040, A000045, A005478, A245306, A335568, A338762, A338794 (indices of the 0's).

a:= proc(n) local i, F, m, t; F, m, t:=
      [1, 2], 0, (<<0|1>, <1|1>>^n)[2, 1]^2+1;
      for i from 3 while F[2]<=t do if isprime(F[2]) and
        irem(t, F[2])=0 then m:=i fi; F:= [F[2], F[1]+F[2]]
      od; numtheory[pi](m)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 25 2020

a[n_] := Module[{i, F = {1, 2}, m = 0, t}, t = MatrixPower[{{0, 1}, {1, 1}}, n][[2, 1]]^2 + 1; For[i = 3, F[[2]] <= t, i++, If[PrimeQ[F[[2]]] && Mod[t, F[[2]]] == 0, m = i]; F = {F[[2]], F[[1]] + F[[2]]}]; PrimePi[m]];
Array[a, 100] (* Jean-François Alcover, Dec 01 2020, after Alois P. Heinz *)

If A335568(n) = 0, then a(n) = 0, otherwise a(n) = A000720(A335568(n)).

A339173 Index of record values of A339082.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 21, 27, 41, 45, 81, 129, 135, 357, 429, 431, 447, 507, 567, 569, 2969, 4721, 5385, 9309, 9675, 14429, 25559, 30755, 35997, 37509, 50831, 81837, 104909, 130019, 148089, 201105, 397377, 433779, 590039, 593687, 604709, 931515, 1049895, 1285605, 1636005, 1803057, 1968719, 2904351, 3244367, 3340365

Offset: 1

Chai Wah Wu, Nov 25 2020

nonn

Also index of record values of A335568.

Cf. A000040, A000045, A001605, A005478, A119984, A245306, A335568, A338762, A338794.

A339082(a(n)) = A119984(n).

For n > 1, a(n) = A001605(n+1)-2.

cp's OEIS Frontend

A339461 Number of Fibonacci divisors of n^2 + 1.

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A335568 a(n) is the number m such that F(m) is the greatest prime Fibonacci divisor of F(n)^2 + 1 where F(n) is the n-th Fibonacci number, or 0 if no such prime factor exists.

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Maple

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Formula

A339082 a(n) is the number m such that F(prime(m)) is the greatest prime Fibonacci divisor of F(n)^2 + 1 where F(n) is the n-th Fibonacci number, or 0 if no such prime factor exists.

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Programs

Maple

Mathematica

Formula

A339173 Index of record values of A339082.

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