A022307 -id:A022307 - cp's OEIS frontend

A074691 Squarefree Fibonacci numbers with odd number of prime factors.

2, 3, 5, 13, 89, 233, 610, 987, 1597, 10946, 28657, 514229, 3524578, 9227465, 24157817, 39088169, 63245986, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 7778742049, 20365011074, 365435296162, 591286729879

Offset: 1

Felice Russo, Sep 03 2002

Agrees for a long time with sequence of Fibonacci numbers whose number of divisors is a Fibonacci number.

			610 belongs to the sequence because it has 3 prime factors (2, 5, 61); it also has 8 divisors (1, 2, 5, 10, 61, 122, 305, 610).

Amiram Eldar, Table of n, a(n) for n = 1..575 (terms 1..100 from T. D. Noe)

Subsequence of A061305 (squarefree Fibonacci numbers).
Cf. A000045.
Cf. A022307, A075735.

with(combinat, fibonacci): m2_fib := proc(n); if (numtheory[mobius](fibonacci(n))=-1) then RETURN(fibonacci(n)); fi; end: seq(m2_fib(i), i=1..100);

Select[Fibonacci[Range[80]], MoebiusMu[#] == -1 &] (* Harvey P. Dale, Aug 23 2011 *)

Fibonacci(n) such that mu(Fibonacci(n)) = -1, where mu(n) is the Moebius mu function (A008683).

More terms from Jani Melik, Oct 07 2002

A080345 a(n) is the number of prime factors in Fibonacci(prime(n)); that is, in the Fibonacci number whose index is the n-th prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 1, 2, 4, 2, 3, 2, 2, 2, 2, 1, 1, 3, 4, 2, 4, 4, 2, 2, 3, 3, 2, 2, 4, 2, 4, 4, 2, 5, 3, 4, 3, 2, 3, 3, 4, 2, 2, 3, 4, 2, 4, 4, 4, 3, 2, 3, 5, 4, 2, 1, 7, 5, 4, 3, 3, 2, 2, 4, 3, 4, 1, 1, 5, 5, 1, 3, 5, 3, 2, 3, 4, 3, 4, 6, 1, 3, 4, 3

Offset: 1

T. D. Noe, Feb 16 2003

In all known examples, Fibonacci(prime(n)) is squarefree, in which case a(n) is well-defined, i.e., the number of distinct prime factors equals the total number of prime factors. But if for some n, Fibonacci(prime(n)) has a repeated prime factor, then a(n) is not well-defined. - Jonathan Sondow, Oct 22 2015

			a(12) = 3 because the 12th prime is 37 and Fibonacci(37) = 24157817 = 73 * 149 * 2221 has 3 prime factors. - clarified by _Jonathan Sondow_, Oct 21 2015

Amiram Eldar, Table of n, a(n) for n = 1..222 (terms 1..168 from T. D. Noe)
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A001605, A022307.

Table[Length[FactorInteger[Fibonacci[Prime[n]]]], {n, 60}]
PrimeNu[Fibonacci[Prime[Range[100]]]] (* Harvey P. Dale, Mar 13 2016 *)

a(n) = omega(fibonacci(prime(n))); \\ Michel Marcus, Oct 22 2015

a(n) = A001221(A000045(A000040(n))). - Michel Marcus, Oct 22 2015

A366769 Number of distinct prime divisors of A001045(n) (Jacobsthal numbers).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 3, 1, 4, 1, 4, 3, 3, 1, 6, 3, 2, 3, 5, 2, 6, 1, 4, 4, 2, 4, 8, 2, 2, 3, 6, 2, 6, 1, 6, 5, 3, 2, 9, 2, 6, 5, 6, 2, 6, 4, 7, 4, 5, 3, 11, 1, 2, 5, 6, 5, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 5, 7, 1, 8, 6, 4, 5, 12, 3, 4

Offset: 1

Sean A. Irvine, Oct 21 2023

nonn

			a(12) = 4 because Jacobsthal(12) = 1365 has prime factors {3, 5, 7, 13}.

Amiram Eldar, Table of n, a(n) for n = 1..1122

Cf. A001045, A001221, A022307, A086598, A364818, A366770.

a(n) = omega(Jacobsthal(n)) = A001221(A001045(n)).

A366780 Number of distinct prime divisors of A000073(n) (tribonacci numbers).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 3, 2, 3, 1, 3, 4, 3, 3, 2, 2, 2, 3, 3, 3, 5, 3, 4, 5, 3, 4, 3, 5, 4, 5, 3, 5, 3, 2, 4, 2, 4, 5, 4, 4, 6, 2, 5, 5, 6, 3, 5, 7, 5, 2, 3, 5, 4, 6, 5, 4, 7, 3, 2, 4, 4, 3, 3, 4, 5, 2, 6, 6, 6, 5, 3, 6, 5, 4, 2, 6, 3, 6, 1, 7

Offset: 2

Sean A. Irvine, Oct 22 2023

nonn

			a(8)=2 because the 8th tribonacci number 24 = 2^3*3 has 2 distinct prime factors.

Amiram Eldar, Table of n, a(n) for n = 2..365

Cf. A000073, A001221, A022307, A366781, A366782.

PrimeNu[LinearRecurrence[{1, 1, 1}, {1, 1, 2}, 87]] (* Amiram Eldar, Oct 23 2023 *)

a(n) = A001221(A000073(n)).

A366781 Number of prime divisors of A000073(n) (tribonacci numbers) (counted with multiplicity).

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 4, 3, 4, 1, 2, 6, 3, 3, 8, 5, 2, 3, 4, 4, 3, 6, 4, 3, 4, 4, 3, 8, 4, 5, 11, 6, 4, 4, 10, 5, 5, 5, 9, 4, 2, 4, 2, 6, 5, 4, 11, 11, 2, 6, 7, 9, 3, 5, 9, 6, 2, 3, 5, 8, 12, 5, 11, 12, 4, 2, 4, 6, 3, 3, 6, 6, 2, 10, 7, 8, 7, 5, 12, 7, 4, 2, 6, 4

Offset: 2

Sean A. Irvine, Oct 22 2023

nonn

			a(8)=4 because the 8th tribonacci number 24 = 2^3*3 has 4 prime factors.

Amiram Eldar, Table of n, a(n) for n = 2..365

Cf. A000073, A001222, A022307, A366780, A366782.

PrimeOmega[LinearRecurrence[{1, 1, 1}, {1, 1, 2}, 84]] (* Amiram Eldar, Oct 23 2023 *)

a(n) = A001222(A000073(n)).

A075735 Squarefree Fibonacci numbers with an even number of prime factors (mu(n)=1).

Original entry on oeis.org

1, 1, 21, 34, 55, 377, 4181, 6765, 17711, 121393, 196418, 317811, 1346269, 2178309, 5702887, 102334155, 165580141, 32951280099, 53316291173, 139583862445, 956722026041, 2504730781961, 10610209857723, 308061521170129

Offset: 1

Jani Melik, Oct 07 2002

			21 is a Fibonacci number and 21=3*7, 34 is a Fibonacci numbers and 34=2*17, ...

Amiram Eldar, Table of n, a(n) for n = 1..505 (terms 1..100 from T. D. Noe)

Subsequence of A061305 (squarefree Fibonacci numbers).
Cf. A000045, A030229, A074691.
Cf. A022307, A053409, A072381.

with(combinat, fibonacci): m1_fib := proc(n); if (numtheory[mobius](fibonacci(n))=1) then RETURN(fibonacci(n)); fi; end: seq(m1_fib(i), i=1..100);

A114840 Indices of Fibonacci numbers with 5 distinct prime factors.

Original entry on oeis.org

30, 36, 42, 44, 45, 50, 57, 63, 66, 68, 69, 75, 76, 98, 111, 118, 124, 134, 141, 153, 169, 172, 183, 185, 201, 202, 203, 213, 218, 229, 247, 253, 267, 302, 303, 329, 335, 347, 363, 371, 373, 377, 381, 382, 386, 395, 398, 413, 415, 439, 443, 461

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1)=30 because 30th Fibonacci number consists of 5 distinct prime factors (i.e., 832040 = 2^3 * 5 * 11 * 31 * 61).

Amiram Eldar, Table of n, a(n) for n = 1..116 (terms 1..106 from Max Alekseyev)
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A114823-A114826, A114836-A114841.

Column k=5 of A303217.

n=1;while(n<305,if(omega(fibonacci(n))==5,print1(n,", "));n++)

{n: A022307(n)=5}. - R. J. Mathar, Nov 29 2015

More terms from Ryan Propper, Apr 26 2006

a(56)-a(106) from Max Alekseyev, Aug 18 2013

A335002 Integers m such that omega(F(m)) = omega(L(m)) where omega is the number of distinct prime divisors function, F(n) and L(n) are the n-th Fibonacci and n-th Lucas numbers.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 17, 18, 21, 26, 33, 42, 46, 47, 55, 58, 66, 69, 73, 77, 85, 89, 93, 102, 103, 107, 111, 117, 121, 123, 132, 139, 167, 171, 177, 179, 181, 187, 201, 205, 207, 213, 219, 221, 233, 241, 246, 247, 253, 257, 262, 267, 269, 273, 279, 281, 282, 293, 295

Offset: 1

Michel Marcus, May 19 2020

nonn

Numbers m such that A022307(m) = A086598(m).

Amiram Eldar, Table of n, a(n) for n = 1..134
Prapanpong Pongsriiam, Fibonacci and Lucas Numbers which have Exactly Three Prime Factors and Some Unique Properties of F18 and L18, Fibonacci Quart. 57 (2019), no. 5, 130-144.

Cf. A001221, A000032, A000045, A022307, A086598, A335001.

lucas(n) = fibonacci(n+1)+fibonacci(n-1);
isok(m) = omega(fibonacci(m))==omega(lucas(m));

A095224 Least squarefree Fibonacci number with exactly n prime divisors.

Original entry on oeis.org

1, 2, 21, 610, 6765, 701408733, 102334155, 190392490709135, 251728825683549488150424261, 23416728348467685, 13598018856492162040239554477268290, 81055900096023504197206408605

Offset: 0

Amarnath Murthy, Jun 10 2004

nonn

Conjecture: The sequence is infinite.
Based on a table of the first 300 Fibonacci numbers, factorized.
Number of prime divisors counted with multiplicity. - Harvey P. Dale, Mar 06 2024

			a(5) = 701408733 = 3 * 43 * 89 * 199 * 307.

Amiram Eldar, Table of n, a(n) for n = 0..24

Cf. A022307, A038575. - R. J. Mathar, Oct 14 2010

From R. J. Mathar, Oct 14 2010: (Start)
A001221 := proc(n) nops(numtheory[factorset](n)) ; end proc:
A095224 := proc(n) for i from 1 do f := combinat[fibonacci](i) ; if A001221(f) =n and numtheory[bigomega](f) = n then return f ; fi; od ; end proc:
for n from 1 do printf("%d,\n",A095224(n)) ; end do: (End)

Table[SelectFirst[{#,PrimeOmega[#]}&/@Select[Fibonacci[Range[200]],SquareFreeQ],#[[2]] == n&],{n,0,11}][[;;,1]] (* Harvey P. Dale, Mar 06 2024 *)

min{A000045(i): A038575(i) = A022307(i) = n}. a(n) >= A060319(n). - R. J. Mathar, Oct 14 2010

a(9) corrected and 3 terms added by R. J. Mathar, Oct 14 2010

A114305 Numbers k such that Fibonacci(k) has more distinct prime factors than k does.

Original entry on oeis.org

8, 9, 15, 16, 18, 19, 20, 21, 24, 25, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Shyam Sunder Gupta, Feb 05 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1377

Cf. A000045, A001221, A022307.

Select[Range[100],PrimeNu[Fibonacci[#]]>PrimeNu[#]&] (* Harvey P. Dale, Oct 25 2011 *)

cp's OEIS Frontend

A074691 Squarefree Fibonacci numbers with odd number of prime factors.

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A080345 a(n) is the number of prime factors in Fibonacci(prime(n)); that is, in the Fibonacci number whose index is the n-th prime.

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A366769 Number of distinct prime divisors of A001045(n) (Jacobsthal numbers).

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A366780 Number of distinct prime divisors of A000073(n) (tribonacci numbers).

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A366781 Number of prime divisors of A000073(n) (tribonacci numbers) (counted with multiplicity).

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A075735 Squarefree Fibonacci numbers with an even number of prime factors (mu(n)=1).

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A114840 Indices of Fibonacci numbers with 5 distinct prime factors.

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A335002 Integers m such that omega(F(m)) = omega(L(m)) where omega is the number of distinct prime divisors function, F(n) and L(n) are the n-th Fibonacci and n-th Lucas numbers.

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