A114842 -id:A114842 - cp's OEIS frontend

A072381 Numbers m such that Fibonacci(m) is a semiprime.

8, 9, 10, 14, 19, 22, 26, 31, 34, 41, 53, 59, 61, 71, 73, 79, 89, 94, 101, 107, 109, 113, 121, 127, 151, 167, 173, 191, 193, 199, 227, 251, 271, 277, 293, 331, 353, 397, 401, 467, 587, 599, 601, 613, 631, 653, 743, 991, 1091, 1223, 1373, 1487

Offset: 1

Shyam Sunder Gupta, Jul 20 2002

Note that there are two cases: (1) n is 2p, in which case the semiprime is Fibonacci(p)*Lucas(p) for some prime p, or (2) n is a power of a prime p^k for k > 0. In the first case, the primes p are in sequence A080327. In the second case, it appears that k=1 except for n = 8, 9 and 121. - T. D. Noe, Sep 23 2005
The associated sequence of Fibonacci numbers contains no squares, since the only Fibonacci numbers which are square are 1 and 144. Consequently this is a subsequence of A114842. - Charles R Greathouse IV, Sep 24 2012
Sequence continues as 1543?, 1709, 1741?, 1759, 1801?, 1889, 1987, ..., where ? marks uncertain terms. - Max Alekseyev, Jul 10 2016

			a(4) = 14 because the 14th Fibonacci number 377 = 13*29 is a semiprime.

Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20.
Ron Knott, Fibonacci numbers
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A053409, A085726 (n such that n-th Lucas number is a semiprime).

Column k=2 of A303215.

Select[Range[200], Plus@@Last/@FactorInteger[Fibonacci[ # ]] == 2&] (Noe)
Select[Range[1500],PrimeOmega[Fibonacci[#]]==2&] (* Harvey P. Dale, Dec 13 2020 *)

for(n=2,9999,bigomega(fibonacci(n))==2&&print1(n",")) \\ - M. F. Hasler, Oct 31 2012

issemi(n)=bigomega(n)==2
is(n)=if(n%2, my(p); if(issquare(n,&p), isprime(p) && isprime(fibonacci(p)) && isprime(fibonacci(n)/fibonacci(p)), isprime(n) && issemi(fibonacci(n))), (isprime(n/2) && isprime(fibonacci(n/2)) && isprime(fibonacci(n)/fibonacci(n/2))) || n==8) \\ Charles R Greathouse IV, Oct 06 2016

More terms from Don Reble, Jul 31 2002
a(49)-a(50) from Max Alekseyev, Aug 18 2013
a(51)-a(52) from Max Alekseyev, Jul 10 2016

A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

3, 8, 4, 15, 9, 5, 20, 16, 10, 6, 30, 24, 18, 12, 7, 40, 36, 27, 21, 14, 11, 70, 48, 42, 28, 33, 19, 13, 60, 81, 54, 44, 32, 35, 22, 17, 80, 72, 104, 56, 45, 52, 37, 25, 23, 90, 84, 110, 105, 64, 50, 55, 38, 26, 29, 140, 126, 88, 112, 136, 78, 57, 74, 39, 31, 43

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
   3,  8, 15, 20, 30,  40,  70,  60,  80,  90, ...
   4,  9, 16, 24, 36,  48,  81,  72,  84, 126, ...
   5, 10, 18, 27, 42,  54, 104, 110,  88, 165, ...
   6, 12, 21, 28, 44,  56, 105, 112,  96, 256, ...
   7, 14, 33, 32, 45,  64, 136, 114, 100, 258, ...
  11, 19, 35, 52, 50,  78, 148, 128, 108, 266, ...
  13, 22, 37, 55, 57,  92, 152, 130, 132, 296, ...
  17, 25, 38, 74, 63,  95, 164, 135, 138, 304, ...
  23, 26, 39, 77, 66,  99, 182, 147, 156, 322, ...
  29, 31, 46, 85, 68, 102, 186, 154, 184, 369, ...

Alois P. Heinz, Antidiagonals n = 1..18, flattened

Columns k=2-16 give: A114842, A114841, A114843, A114840, A114839, A114838, A114837, A114836, A114826, A114825, A114824, A114823, A117529, A117551, A117550.
Row n=1 gives: A060320.
Cf. A000045, A001221, A022307, A303215, A303218.

F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))

nmax = 12; maxIndex = 200;
nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
col[k_] := Select[Range[maxIndex], nu[#] == k&];
T = Array[col, nmax];
A[n_, k_] := T[[k, n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 04 2020 *)

A000045(A(n,k)) = A303218(n,k).

A001221(A000045(A(n,k))) = k.

A227875 Fibonacci numbers which are perfect powers.

Original entry on oeis.org

0, 1, 8, 144

Offset: 1

Jean-François Alcover, Oct 25 2013

Also, Fibonacci numbers which are products of Fibonacci numbers (each greater than 1 when the product is greater than 1 - see A235383). - Rick L. Shepherd, Feb 19 2014
The terms of the subsequence (1, 8, 144) are the Fibonacci numbers that are powerful numbers. - Robert C. Lyons, Jul 12 2016
Also Fibonacci numbers without any primitive divisors. See [Heuberger & Wagner]. - Michel Marcus, Aug 21 2016
It was proved (Bugeaud, Mignotte, and Siksek, 2006, p. 971) that the only perfect powers among the Fibonacci numbers and Lucas numbers are {0, 1, 8, 144} and {1, 4}, respectively. - Daniel Forgues, Apr 09 2018

Vladica Andrejic, On Fibonacci Powers, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 17 (2006), 38-44.
Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20.
Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Annals of Mathematics, 163 (2006), pp. 969-1018.
Clemens Heuberger and Stephan Wagner, On the monoid generated by a Lucas sequence, arXiv:1606.02639 [math.NT], 2016. Gives the complement sequence w.r.t Fibonacci numbers.
J. Mc Laughlin, Small prime powers in the Fibonacci sequence, arXiv:math/0110150 [math.NT] (2001).
Attila Pethő, Diophantine properties of linear recursive sequences II, Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 17:2 (2001), pp. 81-96.

Cf. A000045, A001597, A072381, A114842.

perfectPowerQ[0] = True; perfectPowerQ[1] = True; perfectPowerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1; Union[Select[Fibonacci /@ Range[0, 20], perfectPowerQ]]

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A072381 Numbers m such that Fibonacci(m) is a semiprime.

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A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A227875 Fibonacci numbers which are perfect powers.

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