A114823 - OEIS

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

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OFFSET

1,1

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)

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..54

Blair Kelly, Fibonacci and Lucas Factorizations.

EXAMPLE

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).

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 *)

PROG

(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

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

CROSSREFS