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.

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A083668 Prime indices of prime Fibonacci numbers.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839, 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721
Offset: 1

Views

Author

Cino Hilliard, Jun 14 2003

Keywords

Comments

Same as A001605 without the number 4.
From V. Raman, Oct 04 2012: (Start)
Also the indices of prime Fibonacci numbers which can be written as the sum of two positive squares.
The Fibonacci numbers F(6k+1) and F(6k+5) are congruent to 1 (mod 4).
(End)

Examples

			For Fib(n) to be prime, n must be prime, except for n=4. The first 9 primes are: 2, 3, 5, 7, 11, 13, 17, 19 and 23. The corresponding Fibonacci numbers are: 1, 2, 5, 13, 89, 233, 1597, 4181 and 28657. All of these are prime except Fib(2) = 1 and Fib(19) = 4181. So the first 7 terms of this sequence are 3, 5, 7, 11, 13, 17 and 23.

Crossrefs

Cf. A001605, A075737.

Programs

  • Mathematica
    Do[ If[ PrimeQ[ Fibonacci[ Prime[n]]], Print[ Prime[n]]], {n, 1, 1000}]
  • PARI
    pif(n) = { forprime(x=2,n, if(isprime(fibonacci(x)), print1(x" "))) }
  • PARI
    is(p)=isprime(p) & ispseudoprime(fibonacci(p))  \\ Charles R Greathouse IV, Sep 19 2012

Extensions

More terms from Zak Seidov, Aug 31 2006
Replaced the erroneous example Harry J. Smith, Jan 16 2009
Terms a(42) to a(47) added by V. Raman, Oct 04 2012
Definition and wrong statement in example corrected by M. F. Hasler, Oct 08 2012

A105800 Greatest Fibonacci number that is a proper divisor of the n-th Fibonacci number; a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 5, 1, 8, 1, 13, 5, 21, 1, 34, 1, 55, 13, 89, 1, 144, 5, 233, 34, 377, 1, 610, 1, 987, 89, 1597, 13, 2584, 1, 4181, 233, 6765, 1, 10946, 1, 17711, 610, 28657, 1, 46368, 13, 75025, 1597, 121393, 1, 196418, 89, 317811, 4181, 514229, 1, 832040, 1
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 20 2005

Keywords

Comments

a(A001605(n)) = 1.

Links

Crossrefs

Cf. A000045, A032742, A076984, A139045, A280686, A280688, A280689, A280690, A280698, A280699.
Cf. A046022 (gives the positions of ones).

Programs

  • Mathematica
    nn=70;Join[{1,1},With[{fibs=Fibonacci[Range[nn]]},Table[ Max[ Intersection[ Most[Divisors[fibs[[n]]]],fibs]],{n,3,nn}]]] (* Harvey P. Dale, Apr 10 2012 *)

Formula

From Antti Karttunen, Jan 11 2017: (Start)
a(n) = A280686(A000045(n)).
a(n) = A000045(A032742(n)). [Because A000045 is a divisibility sequence.]
a(A032742(n)) = A280688(n).
(End)

A135956 Members of A050937 (nonprime Fibonacci numbers with prime index) with 5 or more distinct prime factors.

Original entry on oeis.org

322615043836854783580186309282650000354271239929, 1476475227036382503281437027911536541406625644706194668152438732346449273, 22334640661774067356412331900038009953045351020683823507202893507476314037053
Offset: 1

Views

Author

Artur Jasinski, Dec 08 2007

Keywords

Comments

Conjecture: all numbers in this sequence are product of 5 or more sum of two squares

Crossrefs

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135955, A135957.

Programs

  • Mathematica
    k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; Print[n]; If[c > 4, Print[Fibonacci[Prime[n]]]; AppendTo[k, Fibonacci[Prime[n]]]]], {n, 1, 100}]; k

Formula

A050937 INTERSECT { A051270 UNION A074969 UNION ... } = A050937 MINUS {A135955 UNION A135954 UNION A135953}. - R. J. Mathar, Jun 09 2008

Extensions

Edited by R. J. Mathar, Jun 09 2008

A153865 Second-to-largest of four consecutive Fibonacci numbers such that sum of those four consecutive Fibonacci numbers is a prime number.

Original entry on oeis.org

2, 3, 8, 13, 55, 144, 610, 987, 2584, 832040, 14930352, 102334155, 1836311903, 32951280099, 1548008755920, 190392490709135, 8944394323791464, 114059301025943970552219
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Keywords

Comments

1+1+2+3=7, 1+2+3+5=11, 3+5+8+13=29, 5+8+13+21=47, 21+34+55+89=199,...

Crossrefs

Cf. A000045, A001906, A000071, A001605, A153862, A153863

Programs

  • Mathematica
    a=0;b=1;c=1;lst={};Do[d=Fibonacci[n];p=a+b+c+d;If[PrimeQ[p],AppendTo[lst,c]];a=b;b=c;c=d,{n,3,6!}];lst
Select[Partition[Fibonacci[Range[0,200]],4,1],PrimeQ[Total[#]]&][[All,3]] (* Harvey P. Dale, Aug 21 2021 *)

Extensions

Definition modified by Harvey P. Dale, Jul 04 2022

A153867 Primes which are the sum of four consecutive Fibonacci numbers.

Original entry on oeis.org

7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149, 412670427844921037470771, 258899611203303418721656157249445530046830073044201152332257717521
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Keywords

Comments

1+1+2+3=7, 1+2+3+5=11, 3+5+8+13=29, 5+8+13+21=47, 21+34+55+89=199,...

Links

Crossrefs

Cf. A001906, A000071, A001605, A153862, A153863, A153865, A153866.
Primes >3 of A000032 and of A000204.

Programs

  • Mathematica
    a=0;b=1;c=1;lst={};Do[d=Fibonacci[n];p=a+b+c+d;If[PrimeQ[p],AppendTo[lst,p]];a=b;b=c;c=d,{n,3,6!}];lst
Select[Total/@Partition[Fibonacci[Range[500]],4,1],PrimeQ] (* Harvey P. Dale, Apr 23 2014 *)
  • PARI
    list(lim)=my(v=List(),a=7,b=11); while(a<=lim, if(isprime(a), listput(v,a)); [a,b]=[b,a+b]); Vec(v) \\ Charles R Greathouse IV, Oct 07 2016

Formula

a(n) = A005479(n+2). - R. J. Mathar, Jan 05 2009 [See also link to the SeqFan list. - M. F. Hasler, Apr 24 2014]

Extensions

One additional term (a(19)) from Harvey P. Dale, Apr 23 2014

A278245 Least number with the same prime signature as the n-th Fibonacci number: a(n) = A046523(A000045(n)).

Original entry on oeis.org

1, 1, 2, 2, 2, 8, 2, 6, 6, 6, 2, 144, 2, 6, 30, 30, 2, 120, 6, 210, 30, 6, 2, 10080, 12, 6, 210, 210, 2, 9240, 6, 210, 30, 6, 30, 166320, 30, 30, 30, 30030, 6, 9240, 2, 2310, 2310, 30, 2, 2882880, 30, 4620, 30, 210, 6, 120120, 210, 60060, 2310, 30, 6, 232792560, 6, 30, 2310, 30030, 30, 9240, 30, 2310, 2310, 510510, 6, 1396755360, 6, 210, 4620, 2310, 210, 120120, 6
Offset: 1

Views

Author

Antti Karttunen, Nov 16 2016

Keywords

Comments

This sequence can be used as a filter for certain sequences involving Fibonacci numbers as it matches to any sequence that is obtained as f(A000045(n)), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ...").
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

Examples

			From _Michael De Vlieger_, May 18 2017: (Start)
a(6) = 8 because Fibonacci(6) = 8, the multiplicity of the prime factor of 8 is 3; the smallest p^3 = 2^3 = 8.
a(7) = 2 because Fibonacci(7) = 13, the multiplicity of the prime factor of 13 is 1; the smallest p^1 = 2^1 = 2.
a(15) = 30 because Fibonacci(15) = 610. The multiplicities of the prime factors of 610, in order from greatest to least, are {1, 1, 1}, the smallest prime power product p^1 * q^1 * r^1 = 2 * 3 * 5 = 30.
a(18) = 120 because Fibonacci(18) = 2584 = 2^3 * 17 * 19 -> 2^3 * 3 * 5 = 120. (End)

Links

Crossrefs

Cf. A000045, A046523, A278241, A278248.
Cf. A286545 (rgs-version of this sequence), A286467.
Cf. A001605 (positions of 2's), A072381 (of 6's).
Sequences with matching equivalence classes: A063375, A105307, A152774.

Programs

  • Mathematica
    Table[If[# == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &,
Sort[FactorInteger[#][[All, -1]], Greater]]] &@ Fibonacci@ n, {n, 79}] (* Michael De Vlieger, May 18 2017 *)
  • PARI
    A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
f0 = 0; f1 = 1; for(n=1, 10000, write("b278245.txt", n, " ", A046523(f1)); old_f0 = f0; f0 = f1; f1 = f1 + old_f0; );
  • Scheme
    (define (A278245 n) (A046523 (A000045 n)))

Formula

a(n) = A046523(A000045(n)).

A135954 Nonprime Fibonacci numbers with prime indices (A050937) that have exactly 3 prime factors.

Original entry on oeis.org

24157817, 44945570212853, 1500520536206896083277, 50095301248058391139327916261, 11463113765491467695340528626429782121, 30010821454963453907530667147829489881, 2211236406303914545699412969744873993387956988653, 103881042195729914708510518382775401680142036775841
Offset: 1

Views

Author

Artur Jasinski, Dec 08 2007

Keywords

Comments

Conjecture: All numbers in this sequence are products of three sums of two squares.

Crossrefs

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135955, A135956, A135957.

Programs

  • Mathematica
    k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; If[c == 3, AppendTo[k, Fibonacci[Prime[n]]]]], {n, 1, 50}]; k
  • PARI
    f(n) = forprime(x=2, n, p=fibonacci(x); if(!isprime(p) && omega(p) == 3, print1(p", "))) \\ Georg Fischer, Feb 15 2025

Extensions

a(6)-a(8) from Georg Fischer, Feb 15 2025

A135955 (Nonprime Fibonacci numbers with prime indices, A050937) which have exactly 4 prime factors.

Original entry on oeis.org

83621143489848422977, 6161314747715278029583501626149, 289450641941273985495088042104137, 5193981023518027157495786850488117, 66233869353085486281758142155705206899077
Offset: 1

Views

Author

Artur Jasinski, Dec 08 2007

Keywords

Comments

Conjecture: All numbers in this sequence are products of four sums of two squares.

Crossrefs

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135956, A135957.

Programs

  • Mathematica
    k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; If[c == 4, AppendTo[k, Fibonacci[Prime[n]]]]], {n, 1, 50}]; k

A153866 Largest of four consecutive Fibonacci numbers whose sum is a prime number.

Original entry on oeis.org

3, 5, 13, 21, 89, 233, 987, 1597, 4181, 1346269, 24157817, 165580141, 2971215073, 53316291173, 2504730781961, 308061521170129, 14472334024676221, 184551825793033096366333
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Keywords

Comments

1+1+2+3=7, 1+2+3+5=11, 3+5+8+13=29, 5+8+13+21=47, 21+34+55+89=199, ...

Crossrefs

Cf. A000045, A001906, A000071, A001605, A153862, A153863, A153865.

Programs

  • Mathematica
    a=0;b=1;c=1;lst={};Do[d=Fibonacci[n];p=a+b+c+d;If[PrimeQ[p],AppendTo[lst,d]];a=b;b=c;c=d,{n,3,6!}];lst
Alternate: Transpose[Select[Partition[Fibonacci[Range[200]],4,1],PrimeQ[Plus@@#]&]][[4]] (* Harvey P. Dale, Dec 05 2010 *)

A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 5, 2, 3, 4, 4, 2, 5, 2, 5, 4, 3, 2, 7, 3, 3, 4, 5, 2, 7, 2, 5, 4, 3, 4, 8, 2, 3, 4, 7, 2, 7, 2, 5, 6, 3, 2, 9, 3, 5, 4, 5, 2, 7, 4, 7, 4, 3, 2, 11, 2, 3, 6, 6, 4, 7, 2, 5, 4, 7, 2, 11, 2, 3, 6, 5, 4, 7, 2, 9, 5, 3, 2, 11, 4, 3, 4, 7, 2, 11, 4, 5, 4, 3, 4, 11, 2, 5, 6, 8, 2, 7, 2
Offset: 1

Views

Author

Amarnath Murthy, Oct 25 2002

Keywords

Comments

a(A001605(n)) = 2; a(A105802(n)) = n.
It is well known that if k is a divisor of n then F(k) divides F(n). Hence if n has d divisors, one expects that a(n)=d. However because F(1)=F(2)=1, there is one fewer Fibonacci divisor when n is even. So for even n, a(n)=d-1. - T. D. Noe, Jan 18 2006

Examples

			n=12, A000045(12)=144: 5 of the 15 divisors of 144 are also Fibonacci numbers, a(12) = #{1, 2, 3, 8, 144} = 5.

Links

Crossrefs

Cf. A000005, A063375, A000045, A105800.
Cf. A076985.
Cf. A068499.

Programs

  • Maple
    with(combinat, fibonacci):a[1] := 1:for i from 2 to 229 do s := 0:for j from 2 to i do if((fibonacci(i) mod fibonacci(j))=0) then s := s+1:fi:od:a[i] := s:od:seq(a[l],l=2..229);
  • Mathematica
    Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 100}] (Noe)
  • PARI
    {a(n)=if(n<1, 0, numdiv(n)+n%2-1)} /* Michael Somos, Sep 03 2006 */
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n,d, d!=2))} /* Michael Somos, Sep 03 2006 */

Formula

a(n) = A023645(n) + 1. - T. D. Noe, Jan 18 2006
a(n) = tau(n) - [n is even] = A000005(n) - A059841(n). Proof: gcd(Fib(m), Fib(n)) = Fib(gcd(m, n)) and Fib(2) = 1. - Olivier Wittenberg, following a conjecture of Ralf Stephan, Sep 28 2004
The number of divisors of n excluding 2.
a(2n) = A066660(n). a(2n-1) = A099774(n). - Michael Somos, Sep 03 2006
a(3*2^(Prime(n-1)-1)) = 2n + 1 for n > 3. a(3*2^A068499[n]) = 2n + 1, where A068499(n) = {1,2,3,4,6,10,12,16,18,...}. - Alexander Adamchuk, Sep 15 2006

Extensions

Corrected and extended by Sascha Kurz, Jan 26 2003
Edited by N. J. A. Sloane, Sep 14 2006. Some of the comments and formulas may need to be adjusted to reflect the new offset.
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