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.

Previous Showing 11-20 of 20 results.

A135952 Prime factors of composite Fibonacci numbers with prime indices (cf. A050937).

Original entry on oeis.org

37, 73, 113, 149, 157, 193, 269, 277, 313, 353, 389, 397, 457, 557, 613, 673, 677, 733, 757, 877, 953, 977, 997, 1069, 1093, 1153, 1213, 1237, 1453, 1657, 1753, 1873, 1877, 1933, 1949, 1993, 2017, 2137, 2221, 2237, 2309, 2333, 2417, 2473, 2557, 2593, 2749, 2777, 2789, 2797, 2857, 2909, 2917, 3217, 3253, 3313, 3517, 3557, 3733, 4013, 4057, 4177, 4273, 4349, 4357, 4513, 4637, 4733, 4909, 4933
Offset: 1

Views

Author

Artur Jasinski, Dec 08 2007

Keywords

Comments

All numbers in this sequence are congruent to 1 mod 4. - Max Alekseyev.
If Fibonacci(n) is divisible by a prime p of the form 4k+3 then n is even. To prove this statement it is enough to show that (1+sqrt(5))/(1-sqrt(5)) is never a square modulo such p (which is a straightforward exercise).
The n-th prime p is an element of this sequence iff A001602(n) is prime and A051694(n)=A000045(A001602(n))>p. - Max Alekseyev

Links

Crossrefs

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852.

Programs

  • Mathematica
    a = {}; k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], s = FactorInteger[Fibonacci[Prime[n]]]; c = Length[s]; Do[AppendTo[k, s[[m]][[1]]], {m, 1, c}]], {n, 2, 60}]; Union[k]

Extensions

Edited, corrected and extended by Max Alekseyev, Dec 12 2007

A134790 Floor(prime Fibonacci(Prime(k))/Prime(k)).

Original entry on oeis.org

0, 1, 1, 8, 17, 93, 1245, 17732, 10081265, 63217341, 1195118711985005, 8140003186959622868813528, 139669360584622467747806013, 1324290912910022899017738237233285189213652972190967113265372469016533360
Offset: 1

Views

Author

Artur Jasinski, Nov 12 2007

Keywords

Examples

			17732 is in the sequence because floor(514229/29) = 17732, where 29 is the 10th prime number and 514229 = Fibonacci(29) is also a prime. - _Bruno Berselli_, Jul 10 2012

Links

Crossrefs

Cf. A050937, A090819, A134787, A134788, A134789.
Cf. A000040, A000045, A075737.

Programs

  • Mathematica
    k = {}; Do[If[PrimeQ[Fibonacci[Prime[n]]], AppendTo[k, Floor[Fibonacci[Prime[n]]/Prime[n]]]], {n, 1, 100}]; k

A135968 Sum of the distinct prime factors of the Fibonacci number A050937(n).

Original entry on oeis.org

0, 150, 2974, 2443, 62158, 55946694, 2710261050, 555008010, 1547031, 46165377746, 95396546, 92180471494910, 1665088321801550, 364125780, 771601497990, 518283023, 8242065051309594, 32530503217194, 272602401466814027806, 5568053048227732238014, 85526725052226871
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Examples

			a(2) = 150 = 37+113 because A050937(2) = 4181 = 37*113.
a(3) = 2974 = 557 + 2417 because A050937(3) = 1346269 = 557*2417.

Links

Crossrefs

Cf. A000045, A001605, A008472, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135955, A135956, A135957, A135958, A135959, A135969.

Programs

  • Mathematica
    k = {}; Do[If[ ! PrimeQ[Fibonacci[Prime[n]]], b = FactorInteger[Fibonacci[Prime[n]]]; c =Length[FactorInteger[b]]; d = 0; Do[d = d + b[[r]][[1]], {r, 1, c}]; AppendTo[k, d]], {n, 1, 50}]; k

Formula

a(n) = A008472(A050937(n)). - R. J. Mathar, Dec 12 2007

Extensions

Edited by R. J. Mathar, Dec 12 2007
a(19)-a(21) from Amiram Eldar, Oct 13 2024

A134791 a(n) = floor(log(Fibonacci(prime(k))/prime(k))), where k = A119984(n).

Original entry on oeis.org

-1, 0, 0, 2, 2, 4, 7, 9, 16, 17, 34, 57, 60, 166, 200, 201, 209, 237, 266, 267, 1420, 2263, 2582, 4470, 4646, 6933, 12289, 14789, 17311, 18039, 24449, 39369, 50472, 62555, 71250, 96762, 191209, 208726, 283920, 285676, 290979, 448242, 505208, 618634, 787250, 867638
Offset: 1

Views

Author

Artur Jasinski, Nov 12 2007

Keywords

Links

Crossrefs

Cf. A000045, A050937, A090819, A119984, A134787, A134788, A134789, A134790, A134792.

Programs

  • Mathematica
    k = {}; Do[If[PrimeQ[Fibonacci[Prime[n]]], AppendTo[k, Floor[Log[Fibonacci[Prime[n]]/Prime[n]]]]], {n, 1, 200}]; k

Extensions

a(21)-a(46) from Amiram Eldar, Oct 13 2024

A134792 a(n) = round(log(Fibonacci(prime(k))/prime(k))), where k = A119984(n).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 7, 10, 16, 18, 35, 57, 60, 166, 201, 201, 209, 238, 267, 268, 1421, 2263, 2583, 4471, 4647, 6934, 12289, 14789, 17312, 18039, 24450, 39370, 50472, 62555, 71250, 96762, 191210, 208727, 283921, 285676, 290980, 448242, 505208, 618634, 787251, 867638
Offset: 1

Views

Author

Artur Jasinski, Nov 12 2007

Keywords

Links

Crossrefs

Cf. A000045, A050937, A090819, A119984, A134787, A134788, A134789, A134790, A134791.

Programs

  • Mathematica
    k = {}; Do[If[PrimeQ[Fibonacci[Prime[n]]], AppendTo[k, Round[Log[Fibonacci[Prime[n]]/Prime[n]]]]], {n, 1, 200}]; k

Extensions

a(21)-a(46) from Amiram Eldar, Oct 13 2024

A135969 Sum of the prime factors of A135953(n).

Original entry on oeis.org

150, 2974, 62158, 55946694, 2710261050, 555008010, 46165377746, 95396546, 92180471494910, 1665088321801550, 771601497990, 8242065051309594, 32530503217194, 272602401466814027806, 5568053048227732238014, 2811666624525811646469921614, 1966344318693363713266514
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Comments

A subsequence of A135968.

Examples

			a(1) = 150 = 37+113 because A135953(1) = 4181 = 37*113.
a(2) = 2974 = 557+2417 because A135953(2) = 1346269 = 557*2417.

Crossrefs

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135955, A135956, A135957, A135958, A135959, A135968.

Programs

  • Mathematica
    k = {}; Do[If[ ! PrimeQ[Fibonacci[Prime[n]]], b = FactorInteger[Fibonacci[Prime[n]]]; c = Length[FactorInteger[b]]; If[c == 2, d = b[[1]][[1]] + b[[2]][[1]]; AppendTo[k, d]]], {n, 1, 50}]; k
Total[Transpose[FactorInteger[#]][[1]]]&/@Select[Fibonacci[Prime[ Range[50]]], PrimeOmega[#]==2&] (* Harvey P. Dale, Feb 18 2012 *)

Formula

a(n) = A008472(A135953(n)). - R. J. Mathar, Nov 26 2008

Extensions

Definition clarified by R. J. Mathar, Nov 26 2008
a(16)-a(17) from Amiram Eldar, Oct 23 2024

A134850 Number of primes between A075737(n) and A075737(n+1), including one bound.

Original entry on oeis.org

2, 3, 18, 27, 200, 2870, 39492, 22980943, 120106923, 2602986018837012
Offset: 1

Views

Author

Artur Jasinski, Nov 13 2007

Keywords

Links

Crossrefs

Cf. A000045, A050937, A075737, A090819, A134787, A307499.

Programs

  • Mathematica
    a = {}; k = {}; Do[If[PrimeQ[Fibonacci[Prime[n]]], AppendTo[k, Fibonacci[Prime[n]]]], {n, 1, 100}];Do[AppendTo[a, PrimePi[k[[n + 1]]] - PrimePi[k[[n]]]], {n, 1, 9}]; a

Extensions

Edited by N. J. A. Sloane, Oct 07 2008
a(10) using Kim Walisch's primecount, from Amiram Eldar, May 14 2023

A135724 Fibonacci numbers whose indices are prime Fibonacci numbers: a(n) = Fibonacci(A001605(n)).

Original entry on oeis.org

1, 2, 5, 233, 1779979416004714189, 2211236406303914545699412969744873993387956988653
Offset: 1

Views

Author

Artur Jasinski, Nov 26 2007

Keywords

Comments

The only known prime numbers in this sequence are 2, 5 and 233.
The next term (a(6)) has 334 digits. - Harvey P. Dale, Apr 28 2023

Links

Crossrefs

Cf. A000045, A001605, A135723, A134787, A134788, A134789, A134790.

Programs

  • Mathematica
    a = {}; b = {}; Do[If[PrimeQ[c = Fibonacci[n]], w = Fibonacci[Fibonacci[n]]; AppendTo[a, w]; AppendTo[b, n]], {n, 1, 31}]; a
Fibonacci[#]&/@Select[Fibonacci[Range[20]],PrimeQ] (* Harvey P. Dale, Apr 28 2023 *)

Extensions

Corrected by Harvey P. Dale, Apr 28 2023

A135960 Indices where records occur in A134852.

Original entry on oeis.org

1, 2, 4, 14, 37, 59, 144, 173
Offset: 1

Views

Author

Artur Jasinski, Dec 08 2007

Keywords

Crossrefs

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135955, A135956, A135957, A135958, A135959.

Extensions

a(7)-a(8) from Amiram Eldar, Sep 01 2019 (calculated from the b-file at A134852)

A135970 Fibonacci(Mersenne primes): a(n) = Fibonacci(A000668(n)).

Original entry on oeis.org

2, 13, 1346269, 155576970220531065681649693
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Comments

This sequence is a subsequence of A101342. The next term, a(5), has 1712 digits.

Crossrefs

Cf. A000045, A000668, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135955, A135956, A135957, A135958, A135959, A135968, A135969, A101342.

Programs

  • Mathematica
    k = {}; Do[If[PrimeQ[2^n - 1], AppendTo[k, Fibonacci[2^n - 1]]], {n, 1, 15}]; k
Fibonacci/@Select[Table[2^Prime[n]-1,{n,5}],PrimeQ] (* Harvey P. Dale, May 06 2018 *)
Previous Showing 11-20 of 20 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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