A139044 -id:A139044 - cp's OEIS frontend

A060383 a(1) = a(2) = 1; for n >2, a(n) = smallest prime factor of n-th Fibonacci number.

1, 1, 2, 3, 5, 2, 13, 3, 2, 5, 89, 2, 233, 13, 2, 3, 1597, 2, 37, 3, 2, 89, 28657, 2, 5, 233, 2, 3, 514229, 2, 557, 3, 2, 1597, 5, 2, 73, 37, 2, 3, 2789, 2, 433494437, 3, 2, 139, 2971215073, 2, 13, 5, 2, 3, 953, 2, 5, 3, 2, 59, 353, 2, 4513, 557, 2, 3, 5, 2, 269, 3, 2, 5

Offset: 1

Labos Elemer, Apr 03 2001

nonn

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

Tyler Busby, Table of n, a(n) for n = 1..1452 (terms 1..1000 from Alois P. Heinz, derived from Kelly's data, terms 1001..1408 from Amiram Eldar)
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A139044.

[1,1] cat [Minimum(PrimeDivisors(Fibonacci(n))): n in [3..70]]; // Vincenzo Librandi, Dec 25 2016

f[n_] := (FactorInteger@ Fibonacci@ n)[[1,1]]; Array[f, 70] (* Robert G. Wilson v, Jul 07 2007 *)

a(n) = if ((f=fibonacci(n))==1, 1, factor(f)[1,1]); \\ Michel Marcus, Nov 15 2014

a(n) = A020639(A000045(n)). - Michel Marcus, Nov 15 2014

Better definition from Omar E. Pol, Apr 25 2008

A280104 a(n) = smallest prime factor of n-th Lucas number A000032(n), or 1 if there are none.

Original entry on oeis.org

2, 1, 3, 2, 7, 11, 2, 29, 47, 2, 3, 199, 2, 521, 3, 2, 2207, 3571, 2, 9349, 7, 2, 3, 139, 2, 11, 3, 2, 7, 59, 2, 3010349, 1087, 2, 3, 11, 2, 54018521, 3, 2, 47, 370248451, 2, 6709, 7, 2, 3, 6643838879, 2, 29, 3, 2, 7, 119218851371, 2, 11, 47, 2, 3, 709, 2

Offset: 0

Vincenzo Librandi, Dec 26 2016

nonn

From Robert Israel, Jan 05 2017: (Start)
If m and n are odd, m > 1 and m | n, then a(n) <= a(m).
a(n) = 2 if and only if 3 | n.
a(n) = 3 if and only if n is in A091999.
a(n) is never 5.
a(n) = 7 if and only if n is in A259755.
a(n) = A000032(n) if and only if n is in A001606.
(End)

Robert Israel, Table of n, a(n) for n = 0..1000
Fibonacci and Lucas Factorizations

Cf. A000032, A001606, A020639, A079451 (same for largest prime factor), A091999, A139044, A144293, A259755, A279623.

Column k=2 of A238899 (for n>=2).

[2,1] cat [Minimum(PrimeDivisors(Lucas(n))): n in [2..60]];

lucas:= n -> combinat:-fibonacci(n+1)+combinat:-fibonacci(n-1):
spf:= proc(n) local F;
  F:= remove(hastype,ifactors(n,easy)[2],symbol);
  if F <> [] then return min(seq(f[1],f=F)) fi;
min(numtheory:-factorsec(n))
end proc:
spf(1):= 1:
map(spf @ lucas, [$0..200]); # Robert Israel, Jan 05 2017

f[n_]:=(FactorInteger@LucasL@n)[[1, 1]]; Array[f, 60, 0]

a000032(n) = fibonacci(n+1)+fibonacci(n-1)
a(n) = if(a000032(n-1)==1, 1, factor(a000032(n-1))[1, 1]) \\ Felix Fröhlich, Dec 26 2016

a(n) = A020639(A000032(n)). - Felix Fröhlich, Dec 26 2016

Offset changed from Bruno Berselli, Dec 27 2016

A152765 Smallest prime divisor of Catalan number A000108(n), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Omar E. Pol, Dec 15 2008, Jan 03 2009

nonn

a(n) <> 2 iff n = 2^k - 1 (A000225). In fact for k>1, a(2^k-1): 5, 3, 3, 7, 3, 3, 7, 3, 3, 3, 3, 3, 3, ..., . (A120275) - Robert G. Wilson v, Nov 14 2015

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000108, A000225, A120275, A120303, A139044, A152761, A152762, A152763.

[Minimum(PrimeDivisors(Catalan(n))): n in [2..100]]; // Vincenzo Librandi, Jan 04 2017

FactorInteger[#][[1,1]]&/@CatalanNumber[Range[2,80]] (* Harvey P. Dale, Oct 08 2014 *)

a(n) = if (n<=1, 1, factor(binomial(2*n, n)/(n+1))[1, 1]); \\ Michel Marcus, Nov 14 2015; corrected Jun 13 2022

A152765(n) = if(n<2,1,my(c=binomial(2*n, n)/(n+1)); forprime(p=2, oo, if(!(c%p),return(p)))); \\ Antti Karttunen, Jan 12 2019

a(n) = A020639(A000108(n)). - Michel Marcus, Nov 14 2015

Terms a(0) = a(1) = 1 prepended and more terms added by Antti Karttunen, Jan 12 2019

A279623 a(n) = smallest prime factor of n-th Bell number, or 1 if there are none.

Original entry on oeis.org

1, 1, 2, 5, 3, 2, 7, 877, 2, 3, 5, 2, 37, 27644437, 2, 5, 241, 2, 7, 271, 2, 3, 3, 2, 3, 13, 2, 47, 5, 2, 3, 11, 2, 5694673, 3, 2, 29, 3, 2, 6353, 3221, 2, 35742549198872617291353508656626642567, 3, 2, 5, 5, 2, 3, 7615441337805454611187, 2

Offset: 0

Vincenzo Librandi, Dec 25 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 0..131 (terms 0..100 from Robert G. Wilson v)

Cf. A000110, A020639, A139044, A144293.

[1,1] cat [Minimum(PrimeDivisors(Bell(n))): n in [2..50]];

f[n_] := (FactorInteger@ BellB@ n)[[1, 1]]; Array[f, 50, 0]

a(n) = A020639(A000110(n)). - Robert Israel, Dec 25 2016

A139227 Array read by rows: row n lists the proper divisors of n-th Fibonacci number A000045(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 1, 3, 7, 1, 2, 17, 1, 5, 11, 1, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 1, 1, 13, 29, 1, 2, 5, 10, 61, 122, 305, 1, 3, 7, 21, 47, 141, 329, 1, 1, 2, 4, 8, 17, 19, 34, 38, 68, 76, 136, 152, 323, 646, 1292

Offset: 3

Omar E. Pol, Apr 28 2008

			Row ....... Array begins
===========================================================
3 ......... 1
4 ......... 1
5 ......... 1
6 ......... 1, 2, 4
7 ......... 1
8 ......... 1, 3, 7
9 ......... 1, 2, 17
10 ........ 1, 5, 11
11 ........ 1
12 ........ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72
13 ........ 1

Cf. A000045, A133021, A139044, A139045.

f[n_] := Most@ Divisors@ Fibonacci@ n; Flatten@ Array[f, 16, 3] (* Robert G. Wilson v *)

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A060383 a(1) = a(2) = 1; for n >2, a(n) = smallest prime factor of n-th Fibonacci number.

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A280104 a(n) = smallest prime factor of n-th Lucas number A000032(n), or 1 if there are none.

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A152765 Smallest prime divisor of Catalan number A000108(n), with a(0) = a(1) = 1.

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Magma

Mathematica

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PARI

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A279623 a(n) = smallest prime factor of n-th Bell number, or 1 if there are none.

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Author

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Magma

Mathematica

Formula

A139227 Array read by rows: row n lists the proper divisors of n-th Fibonacci number A000045(n).

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Examples

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Mathematica