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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A177086 Semiprimes k that divide Fibonacci(k-1).

Original entry on oeis.org

1891, 4181, 8149, 13201, 15251, 17711, 40501, 51841, 64079, 64681, 67861, 68251, 78409, 88601, 88831, 90061, 96049, 97921, 115231, 118441, 145351, 146611, 153781, 191351, 197209, 218791, 219781, 254321, 272611, 302101, 303101
Offset: 1

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Author

Jonathan Vos Post, Dec 09 2010

Keywords

Comments

This is the semiprime (A001358) analog of A045468. Now A045468 has a very simple characterization: it consists of the primes ending in 1 or 9. Can one say anything about the present sequence?

Examples

			46368/23 = 2016 = 2^5 * 3^2 * 7 so (24-1) | Fibonacci(24) but 24 is not semiprime, so is not in the sequence.
a(1) = 1891 = 31 * 61 is not in the sequence because 1891 divides Fibonacci(1891-1) = Fibonacci(1890).
a(21) = 146611 = 271 * 541 because 146611 | Fibonacci(146610).

Links

Crossrefs

Cf. A000040, A000045, A001358, A069106, A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).
Cf. A177745 (semiprimes k that divide Fibonacci(k+1)).

Programs

  • Mathematica
    Select[Range[310000],PrimeOmega[#]==2 && Divisible[Fibonacci[#-1],#]&] (* Harvey P. Dale, May 02 2016 *)

Formula

{k: k is in A001358 and k|A000045(k-1)} = A069106 INTERSECTION A001358.

A177745 Semiprimes k that divide Fibonacci(k+1).

Original entry on oeis.org

323, 377, 3827, 5777, 10877, 11663, 18407, 19043, 23407, 25877, 27323, 34943, 39203, 51983, 53663, 60377, 75077, 86063, 94667, 100127, 113573, 121103, 121393, 161027, 162133, 182513, 195227, 200147, 231703, 240239, 250277, 294527, 306287, 345913, 381923, 429263, 430127, 454607, 500207, 507527, 548627, 569087, 600767, 635627, 636707, 685583, 697883, 736163, 753377, 775207, 828827, 851927, 948433, 983903
Offset: 1

Views

Author

Jonathan Vos Post, Dec 12 2010

Keywords

Comments

Data from T. D. Noe.

Examples

			a(1) = 323 = 17 * 19 because it is semiprime (product of two prime A000040), and 323 divides F(324) = 23041483585524168262220906489642018075101617466780496790573690289968, with dividend 2^4 * 3^5 * 53 * 107 * 109 * 2269 * 3079 * 4373 * 5779 * 19441 * 11128427 * 62650261 * 1828620361 * 6782976947987.

Crossrefs

Cf. A177086, A000045, A001358, A069106, A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).

Programs

  • Mathematica
    With[{semis=Select[Range[1000000],PrimeOmega[#]==2&]},Select[semis, Divisible[Fibonacci[#+1],#]&]] (* Harvey P. Dale, Aug 20 2012 *)

Formula

{k: k is in A001358 and k|A000045(k+1)}.

A245515 a(n) = n*floor(mod((gcd(n, Fibonacci((-1)^n + n))), 1 + n)/n) for n>=2.

Original entry on oeis.org

1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 59, 0, 61, 0, 0, 0, 0, 0, 0, 0, 0, 0, 71, 0, 0, 0, 0, 0, 0, 0, 79, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

José de Jesús Camacho Medina, Jul 24 2014

Keywords

Comments

Sequence with many prime numbers and zeros.
The primes occurring in this sequence are given in A064739. The subsequence of composite numbers starts 1891, 2737, 2834, 4181, 6601, 6721, 8149, 13201, 13981, ... - Joerg Arndt, Nov 19 2017

Examples

			For n=1, a(1)=1; for n=2, a(2)=2.

Links

Programs

  • Magma
    [n*((Gcd(n, Fibonacci((-1)^n+n)) mod (1+n)) div n): n in [1..100]]; // Vincenzo Librandi, Dec 17 2016
  • Maple
    f:= n -> n*floor(modp((igcd(n, combinat:-fibonacci((-1)^n + n))), 1 + n)/n):
seq(f(n), n=1..100); # Robert Israel, Jul 25 2014
  • Mathematica
    Table[n*Floor[Mod[(GCD[n, Fibonacci[(-1)^n + n]]), 1 + n]/n], {n, 1, 1890}]
  • PARI
    a(n) = n*((gcd(n, fibonacci((-1)^n + n)) % (1 + n))\n); \\ Michel Marcus, Jul 25 2014
  • PARI
    a(n)=gcd(n, lift(((Mod([1,1;1,0],n))^(n+(-1)^n))[1,2]))\n*n \\ Charles R Greathouse IV, Jul 25 2014

Formula

a(n) = n*floor(mod((gcd(n, fibonacci((-1)^n + n))), 1 + n)/n) for n>=1.

A121569 a(n) = Fibonacci((prime(n)+3)/2) - 1.

Original entry on oeis.org

1, 2, 4, 12, 20, 54, 88, 232, 986, 1596, 6764, 17710, 28656, 75024, 317810, 1346268, 2178308, 9227464, 24157816, 39088168, 165580140, 433494436, 1836311902, 12586269024, 32951280098, 53316291172, 139583862444, 225851433716
Offset: 2

Views

Author

Alexander Adamchuk, Aug 08 2006

Keywords

Comments

p = Prime[n] divides a(n) for p = {29,89,101,181,229,349,401,461,509,521,541,709,761,769,809,...} = A047650[n] Primes for which golden mean tau is a quadratic residue or Primes of the form x^2+20y^2.

Crossrefs

Cf. A000045, A121567, A121568, A033205, A045468, A064739, A047650.

Programs

  • Mathematica
    Table[Fibonacci[(Prime[n]+3)/2]-1,{n,2,50}]

Formula

a(n) = Fibonacci[ (Prime[n]+3)/2 ] - 1, n>1. a(n) = Sum[ Fibonacci[k], {k,1,(p-1)/2} ], p = Prime[n], n>1.
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