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

1,1

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++)
(Sage)
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

Cf. A000045, A001221, A022307.

Column k=13 of A303217.

Sequence in context: A179232 A056994 A288461 * A069790 A064224 A069674

Adjacent sequences: A114820 A114821 A114822 * A114824 A114825 A114826

Keyword

hard,nonn

Author

Shyam Sunder Gupta, Feb 19 2006

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

Status

approved