A060385 -id:A060385 - cp's OEIS frontend

A271309 Integers k such that A264137(k) < A060385(k).

52, 60, 65, 74, 75, 76, 85, 108, 111, 121, 124, 125, 127, 131, 132, 140, 144, 150, 153, 156, 158, 172, 175, 180, 183, 185, 195, 201, 209, 213, 216, 220, 225, 250, 263, 287, 300, 301, 327, 328, 335, 337, 339, 344, 356, 370, 402, 404, 408, 412, 417, 423, 433, 435

Offset: 1

Altug Alkan, Apr 03 2016

nonn

For all corresponding values of k, A000129(k) is a composite number. In other words, k cannot be a term of A096650.

			52 is a term because A264137(52) = 66923 < A060385(52) = 90481.

Amiram Eldar, Table of n, a(n) for n = 1..74

Cf. A060385, A096650, A264137.

a060385(n) = my(f=factor(fibonacci(n))[, 1]); f[#f];
a000129(n) = ([2, 1; 1, 0]^n)[2, 1];
a264137(n) = my(p=factor(a000129(n))[, 1]); p[#p];
lista(nn) = for(n=3, nn, if(a264137(n) < a060385(n), print1(n, ", ")));

a(37)-a(54) from Amiram Eldar, May 19 2024

A264137 Largest prime factor of the n-th Pell number, A000129(n).

Original entry on oeis.org

2, 5, 3, 29, 7, 13, 17, 197, 41, 5741, 11, 33461, 239, 269, 577, 8297, 199, 179057, 59, 45697, 5741, 982789, 1153, 29201, 33461, 146449, 337, 44560482149, 269, 3272609, 665857, 52734529, 15607, 1800193921, 199, 1101341, 9369319, 4605197, 5521, 1746860020068409

Offset: 2

Jon E. Schoenfield, Dec 29 2015

nonn

First differs from A246556 at n = 17. Since Pell(17) = 1136689 = 137 * 8297, we find that 137 does not divide any earlier Pell number, and hence A246556(17) = 137, but 8297 is also prime, and so a(17) = 8297.

Daniel Suteu and Jon E. Schoenfield, Table of n, a(n) for n = 2..630 (terms a(2)..a(240) from Jon E. Schoenfield)

Cf. A000129, A006530, A060385, A246556.

Table[FactorInteger[Fibonacci[n, 2]][[-1, 1]], {n, 25}] (* Alonso del Arte, Dec 10 2016 *)
FactorInteger[#][[-1,1]]&/@LinearRecurrence[{2,1},{2,5},60] (* Harvey P. Dale, Jun 08 2019 *)

a(n) = vecmax(factor(([2, 1; 1, 0]^n)[2, 1])[,1]); \\ Daniel Suteu, May 26 2022

a(n) = A006530(A000129(n)).

A193615 Second-largest prime factor of the n-th Fibonacci number, if composite, or 1 otherwise.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 5, 1, 3, 1, 13, 5, 7, 1, 17, 37, 11, 13, 89, 1, 7, 5, 233, 53, 29, 1, 31, 557, 47, 89, 1597, 13, 19, 149, 113, 233, 41, 2789, 211, 1, 199, 61, 461, 1, 47, 97, 151, 1597, 521, 953, 109, 661, 281, 797, 19489, 353, 61, 4513

Offset: 3

Charles R Greathouse IV, Jul 31 2011

nonn

For clarification: if the largest prime factor occurs more than once, then that prime factor is selected.

			F(82) = 2789 * 59369 * 370248451, so a(82) = 59369.

Charles R Greathouse IV, Table of n, a(n) for n = 3..1422

Cf. A060385, A061488, A060442.

factors[n_] := Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]; fn[n_] := Module[{fibn = Fibonacci[n]}, If[PrimeQ[fibn], 1, factors[fibn][[-2]]]]; Table[fn[n], {n, 3, 80}]

a(n)=my(f=factor(fibonacci(n)),t=#f[,1]);if(f[t,2]==1,if(t==1,1,f[t-1,1]),f[t,1])

A121170 Largest prime divisor of Fibonacci(5n).

Original entry on oeis.org

5, 11, 61, 41, 3001, 61, 141961, 2161, 109441, 3001, 474541, 2521, 14736206161, 141961, 230686501, 3041, 3415914041, 109441, 67735001, 570601, 8288823481, 474541, 2441738887963981, 20641, 158414167964045700001, 14736206161, 1114769954367361, 12317523121, 349619996930737079890201

Offset: 1

Alexander Adamchuk, Aug 14 2006

nonn

Except for a(1) = 5 all a(n) are congruent to 1 (mod 10) (final digit is 1). Final digit of most prime divisors of F(5n) is 1.

			a(2) = 11 because F(10)= 5 * 11.
a(4) = 41 because F(20)= 3 * 5 * 11 * 41.
a(10) = 3001 because F(50)= 5^2 * 11 * 101 * 151 * 3001.
a(25) = 158414167964045700001 because F(125)= 5^3 * 3001 * 158414167964045700001.

Amiram Eldar, Table of n, a(n) for n = 1..284

Cf. A000045, A060385.

Table[Max[Flatten[FactorInteger[Fibonacci[5n]]]],{n,1,50}]

a(n) = A060385(5*n).

a(27)-a(29) from Amiram Eldar, Aug 01 2024

A271314 Largest prime factor of the n-th Jacobsthal number, A001045(n).

Original entry on oeis.org

3, 5, 11, 7, 43, 17, 19, 31, 683, 13, 2731, 127, 331, 257, 43691, 73, 174763, 41, 5419, 683, 2796203, 241, 4051, 8191, 87211, 127, 3033169, 331, 715827883, 65537, 20857, 131071, 86171, 109, 25781083, 524287, 22366891, 61681, 8831418697, 5419, 2932031007403, 2113, 18837001

Offset: 3

Altug Alkan, Apr 03 2016

nonn

a(22) = 683 is the first repeated term in this sequence. Note that a(n+2) = A129738(n), for n < 20.

			a(6) = 7 because A001045(6) = 21 = 3*7.

Amiram Eldar, Table of n, a(n) for n = 3..1122 (terms 3..988 from Charles R Greathouse IV)

Essentially a combination of A005420 and A002587.

Cf. A001045, A006530, A060385, A129738, A264137.

FactorInteger[#][[-1, 1]] & /@ Take[#, -(Length@ # - 3)] &@ CoefficientList[Series[x/(1 - x - 2 x^2), {x, 0, 45}], x] (* Michael De Vlieger, Apr 04 2016, after Robert G. Wilson v at A001045 *)

a001045(n) = (2^n - (-1)^n) / 3;
a(n) = vecmax(factor(a001045(n))[,1]);

A074214 Integers m such that F(m) and F(2m) have the same largest prime factor where F(k) denotes the k-th Fibonacci number.

Original entry on oeis.org

3, 15, 21, 23, 25, 29, 33, 35, 39, 43, 45, 51, 55, 59, 63, 65, 75, 82, 83, 85, 87, 93, 99, 105, 107, 109, 111, 115, 119, 123, 125, 127, 131, 132, 133, 135, 137, 139, 142, 143, 145, 147, 151, 153, 158, 161, 166, 171, 173, 175, 179, 181, 183, 185, 187, 189, 191

Offset: 1

Benoit Cloitre, Sep 17 2002

nonn

Why are even values rare? (First one is 82.)

			F(15) = 610 = 2*5*61 and F(30) = 832040 = 2^3*5*11*31*61 hence 15 is in the sequence.

Michel Marcus, Table of n, a(n) for n = 1..225

Cf. A000045, A060385.

Select[Range[3,200],FactorInteger[Fibonacci[#]][[-1,1]]==FactorInteger[ Fibonacci[2#]][[-1,1]]&] (* Harvey P. Dale, Sep 04 2018 *)

f(n) = vecmax(factor(fibonacci(n))[,1]); \\ A060385
isok(m) = (m>2) && (f(m) == f(2*m)); \\ Michel Marcus, Feb 18 2021

More terms from Don Reble, Sep 20 2002

A121169 Largest prime divisor of Fibonacci(100*n).

Original entry on oeis.org

570601, 5738108801, 87129547172401, 3160438834174817356001, 158414167964045700001, 87129547172401, 7358192362316341243805801, 7601587101128729489773008667804801, 427694148584338087778220001

Offset: 1

Alexander Adamchuk, Aug 14 2006

nonn

Most prime divisors of Fibonacci(100*n) are congruent to 1 mod 10 (final digit is 1). It appears that a(n) == 1 (mod 100) for all n.

			a(1) = 570601 because F(100) = 3 * 5^2 * 11 * 41 * 101 * 151 * 401 * 3001 * 570601.
a(10) = 9372625568572722938847095612481183137496995522804466421273200001 because F(1000)= 3 * 5^3 * 7 * 11 * 41 * 101 * 151 * 251 * 401 * 2161 * 3001 * 4001 * 570601 * 9125201 * 112128001 * 1353439001 * 5738108801 * 28143378001 * 5465167948001 * 10496059430146001 * 84817574770589638001 * 158414167964045700001 * 9372625568572722938847095612481183137496995522804466421273200001.

Cf. A000045, A060385.

Table[Max[Flatten[FactorInteger[Fibonacci[100n]]]],{n,10}]

a(n) = my(f=factor(fibonacci(100*n))[, 1]); f[#f]; \\ Jinyuan Wang, Mar 17 2020

a(n) = A060385(100*n).

A325627 a(n) is the largest prime factor in A030426(n).

Original entry on oeis.org

2, 5, 13, 89, 233, 1597, 113, 28657, 514229, 2417, 2221, 59369, 433494437, 2971215073, 55945741, 2710260697, 555003497, 1429913, 46165371073, 86020717, 92180471494753, 99194853094755497, 1665088321800481, 361040209, 770857978613, 512119709, 8242065050061761

Offset: 1

Vincenzo Librandi, May 13 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..222
C. L. Stewart, On Divisors of Fermat, Fibonacci, Lucas, and Lehmer Numbers, Proceedings of the London Mathematical Society, Vol. s3-35, No. 3 (1977), pp. 425-447. See p. 430.

Cf. A000045, A005478, A006530, A030426, A060385, A325626.

[Maximum(PrimeDivisors(Fibonacci(NthPrime(n)))): n in [2..35]];

Table[FactorInteger[Fibonacci [Prime[n]]][[-1, 1]], {n, 2, 30}]

From Amiram Eldar, Oct 25 2024: (Start)
a(n) = A006530(A030426(n)).
a(n) = A060385(prime(n+1)).
a(n) > c * prime(n) * log(prime(n)), where c is an effectively computable positive constant (Stewart, 1977). (End)

A115037 A number n is included if largest prime (or 1 if no prime divides) dividing the n-th Fibonacci number is itself a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 17, 23, 29, 43, 46, 47, 58, 69, 83, 86, 129, 131, 137, 141, 166, 262, 274, 332, 359, 411, 431, 433, 449, 509, 569, 571, 718, 862, 866, 898, 1018, 1138, 1142, 1293, 1347, 1436

Offset: 1

Leroy Quet, Feb 26 2006

A001605 is a subsequence. 2036 and 2276 are also terms. - Chai Wah Wu, May 19 2020

			The 12th Fibonacci number is 144. The largest prime dividing 144 is 3 and 3 is the 4th Fibonacci number. So 12 is in the sequence.

Cf. A000045, A001605, A060385.

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(n) = my(f=fibonacci(n)); (f==1) || isfib(vecmax(factor(f)[,1])); \\ Michel Marcus, Sep 06 2019

More terms from Diana L. Mecum, Jun 02 2007
a(28)-a(40) from Michel Marcus, Sep 06 2019
a(41)-a(46) from Chai Wah Wu, May 19 2020

cp's OEIS Frontend

A271309 Integers k such that A264137(k) < A060385(k).

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A264137 Largest prime factor of the n-th Pell number, A000129(n).

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A193615 Second-largest prime factor of the n-th Fibonacci number, if composite, or 1 otherwise.

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A121170 Largest prime divisor of Fibonacci(5n).

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A271314 Largest prime factor of the n-th Jacobsthal number, A001045(n).

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A074214 Integers m such that F(m) and F(2m) have the same largest prime factor where F(k) denotes the k-th Fibonacci number.

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A121169 Largest prime divisor of Fibonacci(100*n).

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