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.

A175026 Fibonacci entry points: a(n) = smallest m such that prime(A075702(n)) divides Fibonacci(m).

Original entry on oeis.org

432, 127, 1426, 10488, 63221, 1328, 11136, 1291186
Offset: 1

Views

Author

Zak Seidov, Nov 03 2009

Keywords

Comments

In all cases, a(n) is one of divisors of (A075702(n)):
{2160,3048,27094,251712,505768,936240,2182656,2582372}/
{432,127,1426,10488,63221,1328,11136,1291186} = {5,24,19,24,8,705,196,2}.
This is used in Mathematica code for faster search.

Examples

			a(1)=432 because A075702(1)=2160=5*432, prime(2160)=19009, and F(432)/19009= 45104130506533126693784341438185160821786395872599778181861900641867287643757057395776.

Crossrefs

Cf. A001177, A001602, A075702.

Programs

  • Mathematica
    s={2160,3048,27094,251712,505768,936240,2182656,2582372};
Do[sk=s[[k]]; dv=Divisors[sk]; i=2; While[Mod[Fibonacci[dvi=dv[[i]]],Prime[sk]]>0,i++ ]; Print[dvi], {k,8}]

Formula

a(n)=A001602(A075702(n)).

A072123 Remainder when Fibonacci(n) is divided by prime(n).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 2, 11, 26, 27, 33, 28, 33, 46, 33, 4, 22, 27, 20, 69, 15, 22, 88, 44, 92, 100, 21, 76, 21, 69, 41, 116, 134, 44, 76, 70, 117, 80, 157, 129, 87, 73, 27, 157, 1, 5, 208, 27, 108, 1, 203, 230, 19, 112, 143, 206, 258, 31, 3, 146, 266, 117, 213, 211, 168
Offset: 1

Views

Author

Randy L. Ekl, Jun 20 2002

Keywords

Comments

For k=0..8, a(10^k) = 1, 26, 55, 5965, 99584, 728618, 2256590, 61329731, 1081853265. - Zak Seidov, Dec 23 2014

Examples

			a(8) = F(8) mod prime(8) = 21 mod 19 = 2.

Links

Crossrefs

Cf. A000040, A000045, A075702 (locations of 0 in this sequence), A121104.

Programs

  • Magma
    [Fibonacci(n) mod NthPrime(n): n in [1..120]]; // Vincenzo Librandi, Nov 19 2015
  • Maple
    seq(combinat[fibonacci](n) mod ithprime(n), n=1..1000); # Robert Israel, Dec 24 2014
  • Mathematica
    Table[Mod[Fibonacci[n],Prime[n]],{n,70}] (* Harvey P. Dale, Jan 25 2011 *)
  • PARI
    fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
a(n)=lift(fibmod(n,prime(n))) \\ Charles R Greathouse IV, Jun 19 2017

Formula

a(n) = F(n) mod prime(n), where F(n) is the n-th Fibonacci number and prime(n) is the n-th prime number.

A270493 Integers n such that the n-th prime divides the n-th Pell number (A000129(n)).

Original entry on oeis.org

3, 10, 45, 1710, 308961, 601929, 732202, 2214702, 7626372, 13976550, 21971144, 27575700, 207268867, 1014593260, 1134331652, 3140421935, 6196934304, 21338685403, 49990179304, 82456321500
Offset: 1

Views

Author

Altug Alkan, Mar 18 2016

Keywords

Comments

For a(5), corresponding Pell number A000129(308961) has 118263 digits.
Any subsequent terms exceed 2*10^11. - Lucas A. Brown, Mar 08 2024

Examples

			10 is a term because A000129(10) = 2378, A000040(10) = 29 and 2378 mod 29 = 0.

Links

Crossrefs

Cf. A000040, A000129, A075702.

Programs

  • Mathematica
    Select[Range[1, 10000], Divisible[Fibonacci[#, 2], Prime[#]] &] (* Vaclav Kotesovec, Mar 18 2016 *)
  • PARI
    a000129(n) = ([2, 1; 1, 0]^n)[2, 1];
for(n=1, 1e10, if(lift(Mod(a000129(n), prime(n))) == 0, print1(n, ", ")));
  • Python
    # See LINKS.

Extensions

a(6)-a(8) from Gheorghe Coserea, Mar 24 2016
a(9)-a(13) from Daniel Suteu, Nov 07 2019
a(14)-a(20) from Lucas A. Brown, Mar 08 2024

A271332 Integers n such that n-th prime divides the n-th golden rectangle number.

Original entry on oeis.org

6, 15, 51, 754, 803, 2160, 3048, 8315, 18549, 27094, 89929, 251712, 505768, 936240, 1617182, 2182656, 2582372, 5116884, 27067121, 77131559
Offset: 1

Views

Author

Altug Alkan, Apr 04 2016

Keywords

Comments

A075702 is a subsequence.
Integers n such that A000040(n) divides A001654(n).

Examples

			6 is a term because A001654(6) = 104 is divisible by A000040(6) = 13.

Crossrefs

Cf. A001654, A075702.

Extensions

a(11)-a(20) from Daniel Suteu, Nov 07 2019

A328784 Integers k such that the k-th prime divides the k-th Lucas number.

Original entry on oeis.org

2, 4, 5, 608, 1221, 60264, 205965, 994856, 69709961, 3140421767
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 30 2019

Keywords

Crossrefs

Cf. A000032, A000040.
Cf. A075702 (analog with Fibonacci).

Programs

  • Magma
    [n: n in [1..100000] | IsZero(Lucas(n) mod NthPrime(n))];
  • Maple
    a:= 1:
b:= 2:
p:= 2:
Res:= NULL:
for n from 2 to 10^6 do
  c:= a+b;
  b:= a;
  a:= c;
  p:= nextprime(p);
  if a mod p = 0 then
    Res:= Res,n;
  fi
od:
Res; # Robert Israel, Oct 30 2019

Extensions

a(8) from Robert Israel, Oct 30 2019
a(9)-a(10) from Daniel Suteu, Nov 07 2019
Showing 1-5 of 5 results.

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