A114823 Indices of Fibonacci numbers with 13 distinct prime factors.

120, 200, 220, 228, 260, 368, 392, 405, 414, 434, 472, 492, 512, 536, 584, 585, 595, 610, 615, 618, 645, 654, 693, 741, 762, 777, 830, 867, 894, 904, 931, 942, 957, 962, 978, 1045, 1066, 1070, 1074, 1102, 1106, 1108, 1147, 1194, 1209, 1266, 1268, 1309, 1310, 1317

Offset: 1

Author

Shyam Sunder Gupta, Feb 19 2006

Keywords

• hard
• nonn

Comments

From Robert Israel, Aug 18 2015: (Start)

Numbers n such that A022307(n) = 13.

If n is in the sequence, then k*n is not in the sequence for k > 1.

This is because A000045(n) divides A000045(k*n) while Carmichael's theorem says A000045(k*n) has at least one primitive prime factor. (End)

Examples

			a(1)=120 because the 120th Fibonacci number consists of 13 distinct prime factors (i.e., 5358359254990966640871840 = 2^5 * 3^2 * 5 * 7 * 11 * 23 * 31 * 41 * 61 * 241 * 2161 * 2521 * 20641).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..54
• Blair Kelly, Fibonacci and Lucas Factorizations.

Crossrefs

Cf. A000045, A001221, A022307.

Column k=13 of A303217.

Programs

Magma

[n: n in [1..3*10^2] |(#(PrimeDivisors(Fibonacci(n)))) eq 13]; // Vincenzo Librandi, Aug 05 2015

Maple

select(t -> nops(numtheory:-factorset(combinat:-fibonacci(t)))=13, [$1..1000]); # Robert Israel, Aug 10 2015

Mathematica

Select[Range[1250], PrimeNu[Fibonacci[#]]==13&] (* Harvey P. Dale, Apr 30 2015 *)

PARI

n=1;while(n<265,if(omega(fibonacci(n))==13,print1(n,", "));n++)

SageMath

for n in range(1,3*10^2):
    if len(prime_factors(fibonacci(n)))==13:
        print(n) # Manfred Scheucher, Aug 04 2015

Extensions

More terms from Ryan Propper, Apr 26 2006

a(36)-a(45) from Max Alekseyev, Aug 18 2013

a(46)-a(50) from Amiram Eldar, Oct 14 2019