A124132 -id:A124132 - cp's OEIS frontend

A124134 Positive integers n such that Fibonacci(n) = a^2 + b^2, where a, b are integers.

1, 2, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 86, 87, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122, 123, 125, 127

Offset: 1

Melvin J. Knight (melknightdr(AT)verizon.net), Nov 30 2006

nonn

All odd numbers are in this sequence, since the Fibonacci number with index 2m+1 is the sum of the squares of the two Fibonacci numbers with indices m and m+1. Those with even indices ultimately depend on certain Lucas numbers being the sum of two squares (see A124132). Joint work with Kevin O'Bryant and Dennis Eichhorn.

Numbers n such that Fibonacci(n) or Fibonacci(n)/2 is a square are only 0, 1, 2, 3, 6, 12. So a and b must be distinct and nonzero for all values of this sequence except 1, 2, 3, 6, 12. - Altug Alkan, May 04 2016

			14 is in the sequence because F_14=377=11^2+16^2.
16 is not in the sequence because F_16=987 is congruent to 3 (mod 4).

Chai Wah Wu, Table of n, a(n) for n = 1..1322 (terms 1..210 from Joerg Arndt)

Cf. A124132, A000045, A000161, A001481.

a124134 n = a124134_list !! (n-1)
a124134_list = filter ((> 0) . a000161 . a000045) [1..]
-- Reinhard Zumkeller, Oct 10 2013

Select[Range@ 128, SquaresR[2, Fibonacci@ #] > 0 &] (* Michael De Vlieger, May 04 2016 *)

for(n=1, 10^6, t=fibonacci(n); s=sqrtint(t); forstep(i=s, 1, -1, if(issquare(t-i*i), print1(n, ", "); break))) \\ Ralf Stephan, Sep 15 2013

is2s(n)={my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1; } \\ see A001481
for(n=1, 10^6, if(is2s(fibonacci(n)), print1(n, ", "))); \\ Joerg Arndt, Sep 15 2013

from itertools import count, islice
from sympy import factorint, fibonacci
def A124134_gen(): # generator of terms
    return filter(lambda n:n & 1 or all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(fibonacci(n)).items()),count(1))
A124134_list = list(islice(A124134_gen(),30)) # Chai Wah Wu, Jun 27 2022

Intersection of A000045 and A001481.

A000161(A000045(a(n))) > 0. - Reinhard Zumkeller, Oct 10 2013

More terms from Ralf Stephan, Sep 15 2013

A215907 Odd numbers n such that the Lucas number L(n) is the sum of two squares.

Original entry on oeis.org

1, 3, 7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 111, 127, 163, 169, 183, 199, 223, 307, 313, 349, 361, 397, 433, 511, 523, 541, 613, 619, 709, 823, 907, 1087, 1123, 1129, 1147, 1213, 1279, 1434

Offset: 1

V. Raman, Aug 26 2012

These Lucas numbers L(n) have no prime factor congruent to 3 mod 4 to an odd power.
Also, numbers n such that L(n) can be written in the form a^2 + 5*b^2.
Subsequence of A124132.
Is this A124132 without the 6? - Joerg Arndt, Sep 07 2012
Any prime factor of Lucas(n) for the prime values of n is always of the form 1 (mod 10) or 9 (mod 10).
A number n can be written in the form a^2 + 5*b^2 if and only if n is 0, or of the form 2^(2i) 5^j Product_{p==1 or 9 mod 20} p^k Product_{q==3 or 7 mod 20) q^(2m) or of the form 2^(2i+1) 5^j Product_{p==1 or 9 mod 20} p^k Product_{q==3 or 7 mod 20) q^(2m+1), for integers i,j,k,m, for primes p,q.
1501 <= a(42) <= 1531. 1531, 1651, 1747, 1849, 1951, 2053, 2413, 2449, 2467, 4069, 5107, 5419, 5851, 7243, 7741, 8467, 13963, 14449, 14887, 15511, 15907, 35449, 51169, 193201, 344293, 387433, 574219, 901657, 1051849 are terms. - Chai Wah Wu, Jul 22 2020

			Lucas(19) = 9349 = 95^2 + 18^2.
Lucas(19) = 9349 = 23^2 + 5*42^2.

Blair Kelly, Fibonacci and Lucas factorizations
Eric W. Weisstein, Sum of squares function

Cf. A000032, A124130, A215809, A215906.
Cf. A180363.
Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).

for(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; has=0; for(j=1, #a, if(a[1, j]%4==3&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==1, print(i", "))) \\ a^2 + b^2 form.

for(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i", "))) \\ a^2 + 5*b^2 form.

17 more terms from V. Raman, Aug 28 2012
A215940 merged into this sequence by T. D. Noe, Sep 21 2012
a(38)-a(41) from Chai Wah Wu, Jul 22 2020

A356809 Fibonacci numbers which are not the sum of two squares.

Original entry on oeis.org

3, 21, 55, 987, 2584, 6765, 17711, 46368, 317811, 832040, 2178309, 5702887, 14930352, 102334155, 267914296, 701408733, 1836311903, 4807526976, 12586269025, 32951280099, 86267571272, 225851433717, 591286729879, 1548008755920, 10610209857723

Offset: 1

Ctibor O. Zizka, Aug 29 2022

nonn

			F(4) = 3; 3 != x^2 + y^2 as no positive integers x, y >= 0 are the solution of this Diophantine equation.

Chai Wah Wu, Table of n, a(n) for n = 1..472

Intersection of A000045 and A022544.

Cf. A001481, A022340, A124132, A124134, A236264.

Select[Fibonacci[Range[65]], SquaresR[2, #] == 0 &] (* Amiram Eldar, Aug 29 2022 *)

is(n)=if(n%4==3, return(1)); my(f=factor(n)); for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(1))); 0; \\ A022544
lista(nn) = select(is, apply(fibonacci, [1..nn])); \\ Michel Marcus, Sep 04 2022

from itertools import islice
from sympy import factorint
def A356809_gen(): # generator of terms
    a, b = 1, 2
    while True:
        if any(p&3==3 and e&1 for p, e in factorint(a).items()):
            yield a
        a, b = b, a+b
A356809_list = list(islice(A356809_gen(),30)) # Chai Wah Wu, Jan 10 2023

A215938 Numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 5*b^2.

Original entry on oeis.org

1, 2, 5, 8, 11, 12, 25, 29, 32, 41, 48, 55, 89, 121, 125, 128, 131, 145, 179, 192, 205, 275, 331, 359, 401, 421, 431, 445, 449, 509, 512, 569, 571, 601, 605, 625, 631, 655, 659, 691, 725, 768, 895, 911, 1025, 1375

Offset: 1

V. Raman, Aug 27 2012

A number n can be written in the form a^2+5*b^2 if and only if n is 0, or of the form 2^(2i) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m) or of the form 2^(2i+1) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m+1), for integers i,j,k,m, for primes p,q.

Blair Kelly, Fibonacci and Lucas factorizations

Cf. A000045, A215939, A124132.

Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).

for(i=2, 500, a=factorint(fibonacci(i))~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i", ")))

Terms corrected by V. Raman, Sep 20 2012

a(46) from Amiram Eldar, Oct 14 2019

A215939 Prime numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 5*b^2.

Original entry on oeis.org

2, 5, 11, 29, 41, 89, 131, 179, 331, 359, 401, 421, 431, 449, 509, 569, 571, 601, 631, 659, 691, 911

Offset: 1

V. Raman, Aug 27 2012

A number n can be written in the form a^2+5*b^2 if and only if n is 0, or of the form 2^(2i) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m) or of the form 2^(2i+1) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m+1), for integers i,j,k,m, for primes p,q.

Blair Kelly, Fibonacci and Lucas factorizations

Cf. A000045, A215938, A124132.

Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).

forprime(i=2, 500, a=factorint(fibonacci(i))~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i", ")))

Terms corrected by V. Raman, Sep 20 2012

A236264 Even indices of Fibonacci numbers which are the sum of two squares.

Original entry on oeis.org

0, 2, 6, 12, 14, 26, 38, 62, 74, 86, 98, 122, 134, 146, 158, 182, 222, 254, 326, 338, 366, 398, 446, 614, 626, 698, 722, 794, 866, 1022, 1046, 1082, 1226, 1238, 1418, 1646, 1814, 2174, 2246, 2258, 2294, 2426, 2558

Offset: 1

Jean-François Alcover, Jan 21 2014

The first 10 such Fibonacci numbers are 0, 1, 8, 144, 377, 121393, 39088169, 4052739537881, 1304969544928657, 420196140727489673.
Ballot & Luca (Proposition 1) show that this sequence has asymptotic density 0. - Charles R Greathouse IV, Jan 21 2014
a(43) >= 2558. Determining this term requires factoring the Lucas number L_1279. - Charles R Greathouse IV, Jan 21 2014
3002 <= a(44) <= 3302. 3302, 3698, 4898 are terms. - Chai Wah Wu, Jul 23 2020

			Fibonacci(14) = 377 = 19^2 + 4^2, so 14 is in the sequence.

Christian Ballot and Florian Luca, On the equation x^2+dy^2=Fn, Acta Arith. 127 (2007), 145-155.
Kevin O'Bryant, Which Fibonacci numbers are the sum of two squares?, MathOverflow.

Cf. A000045, A001481, A124132.

Reap[For[n = 0, n <= 400, n = n+2, If[Reduce[Fibonacci[n] == x^2 + y^2, {x, y}, Integers] =!= False, Print[n]; Sow[n]]]][[2, 1]]

is(n)=if(n%2, return(0)); my(f=factor(fibonacci(n))); for(i=1,#f~, if(f[i,1]%4==3 && f[i,2]%2, return(0))); 1 \\ Charles R Greathouse IV, Jan 21 2014

default(factor_add_primes, 1);
is(n)={
    if(n%2,return(0));
    my(f=fibonacci(n),t);
    if(f%4==3,return(0));
    forprime(p=2,min(log(f)^2,1e5),
        if(f%p==0,
            t=valuation(f,p);
            if(p%4==3&&t%2,return(0));
            f/=p^t;
            if(f%4==3,return(0))
        )
    );
    fordiv(n,d,
        if(d==n, break);
        t=factor(fibonacci(d))[,1];
        for(i=1,#t,
            if(t[i]%4==3 && valuation(f,t[i])%2, return(0));
            f/=t[i]^valuation(f,t[i]);
            if(f%4==3,return(0))
        )
    );
    f=factor(f);
    for(i=1,#f[,1],
        if(f[i,2]%2&&f[i,1]%4==3,return(0))
    );
    1
}; \\ Charles R Greathouse IV, Jan 21 2014

from itertools import count, islice
from sympy import factorint, fibonacci
def A236264_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(fibonacci(n)).items()),count(0,2))
a236264_list = list(islice(A236264_gen(),10)) # Chai Wah Wu, Jun 27 2022

a(n) = 2*A124132(n-1).

a(32)-a(42) from Charles R Greathouse IV, Jan 21 2014

a(43) from Chai Wah Wu, Jul 23 2020

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A124134 Positive integers n such that Fibonacci(n) = a^2 + b^2, where a, b are integers.

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A215907 Odd numbers n such that the Lucas number L(n) is the sum of two squares.

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A356809 Fibonacci numbers which are not the sum of two squares.

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A215938 Numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 5*b^2.

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A215939 Prime numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 5*b^2.

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A236264 Even indices of Fibonacci numbers which are the sum of two squares.

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