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 11-18 of 18 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

A135959 a(n) = smallest Fibonacci number with prime index which has exactly n prime factors.

Original entry on oeis.org

1, 2, 4181, 24157817, 83621143489848422977, 322615043836854783580186309282650000354271239929
Offset: 0

Views

Author

Artur Jasinski, Dec 08 2007

Keywords

Comments

a(6) has 105 digits.
a(7) = 22334640661774067356412331900038009953045351020683823507202893507476314037053.
Variant of A114722. [From R. J. Mathar, Oct 28 2008]

Crossrefs

Cf. A000045, A050937, A075737, A134852, A135953-A135956, A135958.

Formula

a(n) = Fibonacci(A135958(n)).

Extensions

Edited by David Wasserman, Mar 26 2008

A135979 Indices n such that 2^prime(n)-1 has exactly 2 distinct prime factors.

Original entry on oeis.org

5, 9, 12, 13, 17, 19, 23, 25, 26, 27, 29, 32, 33, 34, 35, 39, 45, 46, 49, 53, 57, 58, 60, 62, 69, 74, 75, 82, 88, 93, 99, 129, 140, 152, 164, 166, 168, 178, 179
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Comments

a(40)>=206. - Amiram Eldar, Sep 29 2018

Crossrefs

Cf. A000225, A065341, A054723, A134852, A135975, A135976, A135977, A135978.

Programs

  • Mathematica
    k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, n]]], {n, 1, 40}]; k
Select[Range[40],PrimeNu[2^Prime[#]-1]==2&] (* Harvey P. Dale, Jul 07 2013 *)

Formula

Equals {k: A001221(A001348(k)) = 2}. a(n) = A049084(A135978(n)). - R. J. Mathar, May 03 2008

Extensions

Edited by R. J. Mathar, May 03 2008
a(17)-a(34) from Donovan Johnson, Jun 14 2009
a(35)-a(39) from Amiram Eldar, Sep 29 2018

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

A135980 Numbers k such that the Mersenne number 2^prime(k)-1 is composite.

Original entry on oeis.org

5, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Comments

A135979 is a subsequence of this sequence.

Crossrefs

Cf. A000225, A065341, A054723, A134852, A135975, A135976, A135977, A135978, A135979.

Programs

  • Mathematica
    k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], AppendTo[k, n]], {n, 1, 40}]; k
m = PrimePi @ MersennePrimeExponent @ Range[13]; Complement[Range[m[[-1]]], m] (* Amiram Eldar, Mar 12 2020 *)
  • PARI
    isok(k) = !isprime(2^prime(k)-1); \\ Michel Marcus, Mar 12 2020

Formula

prime(a(n)) = A054723(n).
a(n) = pi(A054723(n)).

Extensions

More terms from Amiram Eldar, Mar 12 2020

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

A135386 Mersenne composites A065341 with 4 or more prime factors.

Original entry on oeis.org

10384593717069655257060992658440191, 2854495385411919762116571938898990272765493247, 182687704666362864775460604089535377456991567871
Offset: 1

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Links

Crossrefs

Subsequence of A065341.
Cf. A000225, A054723, A134852, A135975, A135976, A135977, A135979.

Programs

  • Maple
    A135386 := proc(n) local i;
i := 2^(ithprime(n))-1:
if (nops(numtheory[factorset](i)) > 3) then
   RETURN (i)
fi: end: seq(A135386(n), n=1..37); # Jani Melik, Feb 09 2011
  • Mathematica
    k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d >3, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k

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 11-18 of 18 results.

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