cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A076697 Indices of record values in A079451, largest prime factor of Lucas numbers A000032.

Original entry on oeis.org

0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 26, 31, 37, 41, 47, 53, 61, 68, 71, 76, 79, 86, 113, 136, 164, 172, 178, 202, 218, 229, 262, 278, 284, 307, 313, 328, 353, 373, 436, 443, 458, 487, 503, 557, 577, 586, 613, 617, 746, 751, 758, 863, 914
Offset: 0

Views

Author

Shane Findley, Oct 25 2002

Keywords

Comments

From M. F. Hasler, Apr 09 2025: (Start)
Original name: Next-to-largest factor of Lucas(n).
The offset 0 is coherent with the fact that the initial term is a starting value rather than a record value.
When A000032(n) is prime (<=> n is in A001606), it necessarily sets a new record for the largest prime factor, since A000032 is increasing from the second term on. Therefore, A001606 is a subsequence. (End)

Links

  • Ron Knott, Lucas numbers.
  • Douglas S. McNeil, in reply to Robert Israel, A076697, SeqFan google group, April 9, 2025.

Crossrefs

Cf. A000042 (Lucas numbers, starting with 2), A079451 (largest prime factor of these).
Cf. A001606 (Indices of prime Lucas numbers: a subsequence).

Programs

  • PARI
    A076697_first(n, m=0)=vector(n,i, i>1 || n=-1; until(mA079451(n++), m), );n) \\ M. F. Hasler, Apr 09 2025
  • Python
    def A076697(n):
    try: terms, M = A076697.terms, A076697.M
    except AttributeError: A076697.terms = terms = [0]; A076697.M = M = 2
    while len(terms) <= n: terms.append(next(i for i in range(terms[-1]+1, 1<<59)
        if M < (M:=max(A079451(i),M)))); A076697.M = M
    return terms[n] # M. F. Hasler, Apr 10 2025

Extensions

New definition and data corrected and extended by M. F. Hasler, Apr 09 2025

A121171 Largest prime divisor of Lucas(5*n), where Lucas(k) = A000032(k).

Original entry on oeis.org

11, 41, 31, 2161, 151, 2521, 911, 3041, 541, 570601, 39161, 20641, 24571, 12317523121, 18451, 23725145626561, 12760031, 10783342081, 87382901, 5738108801, 767131, 59996854928656801, 686551, 23735900452321, 28143378001, 42426476041450801, 119611
Offset: 1

Views

Author

Alexander Adamchuk, Aug 14 2006

Keywords

Comments

Final digit of a(n) is 1. Mod[a(n),10] = 1. Final digit of many prime divisors of Lucas(5*n) is 1.

Links

Crossrefs

Cf. A000032, A000045, A079451.

Programs

  • Mathematica
    Table[Max[Flatten[FactorInteger[Fibonacci[5n-1]+Fibonacci[5n+1]]]],{n,1,40}]
  • PARI
    lucas(n) = fibonacci(n+1)+fibonacci(n-1); \\ A000032
a(n) = vecmax(factor(lucas(5*n))[,1]); \\ Daniel Suteu, May 26 2022

Formula

a(n) = A006530(A000032(5*n)) = A079451(5*n). - Daniel Suteu, May 26 2022

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

Views

Author

Vincenzo Librandi, Dec 26 2016

Keywords

Comments

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)

Links

Crossrefs

Cf. A000032, A001606, A020639, A079451 (same for largest prime factor), A091999, A139044, A144293, A259755, A279623.
Column k=2 of A238899 (for n>=2).

Programs

  • Magma
    [2,1] cat [Minimum(PrimeDivisors(Lucas(n))): n in [2..60]];
  • Maple
    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
  • Mathematica
    f[n_]:=(FactorInteger@LucasL@n)[[1, 1]]; Array[f, 60, 0]
  • PARI
    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

Formula

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

Extensions

Offset changed from Bruno Berselli, Dec 27 2016

A121708 Numerator of Sum/Product of first n Fibonacci numbers A000045[n].

Original entry on oeis.org

1, 2, 2, 7, 2, 1, 11, 3, 11, 1, 29, 47, 29, 1, 19, 41, 19, 1, 199, 23, 199, 1, 521, 281, 521, 1, 31, 2207, 31, 1, 3571, 107, 3571, 1, 9349, 2161, 9349, 1, 211, 13201, 211, 1, 64079, 1103, 64079, 1, 15251, 90481, 15251, 1, 5779, 14503, 5779, 1, 1149851, 2521
Offset: 1

Views

Author

Alexander Adamchuk, Aug 16 2006, Sep 21 2006

Keywords

Comments

a(1) = 1 and a(4k+2) = 1 for k>0.
For k >1 a(4k-1) = a(4k+1) = A072183(2k+1) = A061447(2k+1) Primitive part of Lucas(n).

Crossrefs

Cf. A000045, A000032, A079451, A001578, A096362, A058036, A121709, A090585.
Cf. A061447, A072183.

Programs

  • Mathematica
    Table[Numerator[Sum[Fibonacci[k],{k,1,n}]/Product[Fibonacci[k],{k,1,n}]],{n,1,100}]
With[{fibs=Fibonacci[Range[60]]},Numerator[Accumulate[fibs]/Rest[ FoldList[ Times,1,fibs]]]] (* This is significantly faster than the first program above *) (* Harvey P. Dale, Aug 19 2012 *)

Formula

a(n) = numerator( sum(k=1..n, Fibonacci(k)) / prod(k=1..n, Fibonacci(k)) ).

A121709 Numerator of Sum/Product of first n Lucas numbers A000032[n].

Original entry on oeis.org

1, 4, 2, 5, 13, 1, 73, 5, 7, 1, 37, 5, 1361, 1, 223, 25, 4673, 1, 24473, 25, 16019, 1, 83879, 65, 62743, 1, 20533, 65, 1505173, 1, 7881193, 85, 5158309, 1, 27009259, 425, 1400221, 1, 1446283, 2225, 69237359, 1, 51790217, 445, 1660959719, 1, 8696897999
Offset: 1

Views

Author

Alexander Adamchuk, Aug 16 2006

Keywords

Comments

5 divides a(4k). a(1) = 1 and a(4k+2) = 1 for k>0.

Crossrefs

Cf. A000045, A000032, A079451, A001578, A096362, A058036, A121708, A090585.

Programs

  • Mathematica
    Table[Numerator[Sum[Fibonacci[k-1]+Fibonacci[k+1],{k,1,n}]/Product[Fibonacci[k-1]+Fibonacci[k+1],{k,1,n}]],{n,1,100}]

Formula

a(n) = Numerator[Sum[Lucas[k],{k,1,n}]/Product[Lucas[k],{k,1,n}]], where Lucas[k] = Fibonacci[k-1] + Fibonacci[k+1].
Showing 1-5 of 5 results.

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