A338762 -id:A338762 - 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

A338794 Indices k of Fibonacci numbers F(k) such that F(k)^2 + 1 has no Fibonacci prime factor.

Original entry on oeis.org

39, 60, 69, 72, 99, 102, 105, 108, 111, 150, 165, 180, 192, 195, 198, 225, 228, 231, 240, 270, 279, 282, 309, 312, 315, 348, 351, 381, 399, 420, 441, 459, 462, 465, 489, 501, 522, 588, 591, 600, 615, 618, 642, 645, 660, 675, 702, 741, 759, 771, 810, 822, 825, 828

Offset: 1

Michel Lagneau, Nov 09 2020

nonn

Numbers k such that A338762(k) = 0.

			39 is in the sequence because F(39)^2 + 1 = 63245986^2 + 1 = 73*149*2221*2789*59369 with no Fibonacci prime factors.
38 is not in the sequence because F(38)^2 + 1 = 39088169^2 + 1 =  2*73*149*233*2221*135721. The numbers and 2, 233 are Fibonacci prime factors.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000045, A005478, A168063, A245306, A338762.

a:= proc(n) local F, m, t; F, m, t:=
      [1, 2], 0, (<<0|1>, <1|1>>^n)[2, 1]^2+1;
      while F[2]<=t do if isprime(F[2]) and irem(t, F[2])=0
        then m:=F[2] fi; F:= [F[2], F[1]+F[2]]
      od; m
    end:
for n from 1 to 100 do :
if a(n)=0 then printf(`%d, `,n):else fi:
od: # program from Alois P. Heinz, adapted for the sequence. See A338762.

A338762[n_] := Module[{F, m, t}, F = {1, 2}; m = 0; t = MatrixPower[{{0, 1}, {1, 1}}, n][[2, 1]]^2 + 1; While[F[[2]] <= t, If[PrimeQ[F[[2]]] && Mod[t, F[[2]]] == 0, m = F[[2]]]; F = {F[[2]], F[[1]] + F[[2]]}]; m];
Reap[For[k = 1, k <= 1000, k++, If[A338762[k] == 0, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Mar 16 2025, after Alois P. Heinz *)

isok(n) = {my(i=0, f=0, x=fibonacci(n)^2+1, m=0); while(f < x, i++; f = fibonacci(i); if (ispseudoprime(f) && (x%f) == 0, return (0));); return(1);} \\ Michel Marcus, Nov 13 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)).

A352290 Numbers m such that the greatest prime factor of m^2 + 1 is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 18, 34, 55, 57, 89, 123, 144, 233, 239, 322, 377, 411, 500, 568, 610, 746, 788, 843, 987, 1487, 1542, 1568, 1636, 2207, 2584, 2707, 3173, 3639, 3793, 3804, 3817, 4050, 4181, 4217, 4594, 4662, 5270, 5778, 6107, 6613, 8595, 8972, 10341, 10569

Offset: 1

Michel Lagneau, Mar 11 2022

nonn

A281618 is a subsequence.
The corresponding greatest prime Fibonacci factors of the sequence are 2, 5, 5, 13, 5, 13, 13, 89, 89, 13, 233, 89, 233, ...
The Fibonacci numbers of the sequence are 1, 2, 3, 5, 8, 34, 55, 89, 144, 233, 377, 610, 987, 2584, 4181, 10946, 17711, ... (subsequence of A000045).
The Lucas numbers of the sequence are 1, 2, 3, 7, 18, 123, 322, 843, 2207, 5778, 39603, 103682, ... (subsequence of A000032).
The prime numbers of the sequence are 2, 3, 5, 7, 89, 233, 239, 1487, 2207, 2707, 3793, 4217, 11789, 11981, 13763, ... including the prime Fibonacci numbers 2, 3, 5, 89, 233, 1066340417491710595814572169, ... (subsequence of A005478).

			18 is in the sequence because 18^2 + 1 = 5^2*13 and 13 is a Fibonacci number.

Cf. A000032, A000045, A002522, A005478, A014442, A248516, A338762, A339461.

q:= n-> (t-> ormap(issqr, [t+4, t-4]))(5*max(numtheory[factorset](n^2+1))^2):
select(q, [$1..12000])[];  # Alois P. Heinz, Mar 11 2022

With[{f = Fibonacci[Range[21]], m = f[[-1]]}, Select[Range[m], MemberQ[f, FactorInteger[#^2 + 1][[-1, 1]]] &]] (* Amiram Eldar, Mar 11 2022 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(m) = isfib(vecmax(factor(m^2+1)[,1])); \\ Michel Marcus, Mar 11 2022

A360107 Numbers k such that sigma_2(Fibonacci(k)^2 + 1) == 0 (mod Fibonacci(k)).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 19, 21, 25, 27, 31, 41, 45, 49, 81, 85, 129, 133, 135, 139, 357, 361, 429, 431, 433, 435, 447, 451, 507, 511, 567, 569, 571, 573

Offset: 1

Michel Lagneau, Jan 26 2023

			7 is in the sequence because the divisors of Fibonacci(7)^2 + 1 = 13^2 + 1 = 170 are {1, 2, 5, 10, 17, 34, 85, 170}, and 1^2 + 2^2 + 5^2 + 10^2 + 17^2 + 34^2 + 85^2 + 170^2 = 37700 = 13*2900 == 0 (mod 13).

Cf. A000045, A001157, A067719, A245236, A245306, A338762, A360105.

Select[Range[140],Divisible[DivisorSigma[2,Fibonacci[#]^2+1],Fibonacci[#]]&]

isok(k) = my(f=fibonacci(k)); sigma(f^2 + 1, 2) % f == 0; \\ Michel Marcus, Jan 26 2023

a(24)-a(37) from Daniel Suteu, Jan 27 2023

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.

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

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A338794 Indices k of Fibonacci numbers F(k) such that F(k)^2 + 1 has no Fibonacci prime factor.

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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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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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A352290 Numbers m such that the greatest prime factor of m^2 + 1 is a Fibonacci number.

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A360107 Numbers k such that sigma_2(Fibonacci(k)^2 + 1) == 0 (mod Fibonacci(k)).

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A339173 Index of record values of A339082.

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