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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A155011 Sophie Germain Fibonacci prime numbers.

Original entry on oeis.org

2, 3, 5, 89, 233, 10597999265301490732599643671505003412515860435409421932560009680142974347195483140293254396195769876129909
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

Keywords

Comments

2*2+1=5, 2*3+1=7, 2*5+1=11, ...
No additional terms up through Fibonacci(10000). - Harvey P. Dale, Nov 26 2013
No additional terms up through Fibonacci(50000). - Chai Wah Wu, Nov 04 2015

Crossrefs

Cf. A005478, A000045, A005384, A263880.

Programs

  • Mathematica
    a={};Do[f=Fibonacci[n];If[PrimeQ[f],If[PrimeQ[2*f+1],AppendTo[a,f]]],{n,3*6!}];a
  Select[Fibonacci[Range[2000]],And@@PrimeQ[{#,2#+1}]&] (* Harvey P. Dale, Nov 26 2013 *)
  • Python
    from gmpy2 import is_prime
A155011_list = []
a, b, a2, b2 = 0, 1, 1, 3
for _ in range(10**6):
    if is_prime(b) and is_prime(b2):
        A155011_list.append(b)
    a, b, a2, b2 = b, a+b, b2, a2+b2-1 # Chai Wah Wu, Nov 04 2015

Formula

a(n) = (A263880(n) - 1)/2. - Jonathan Sondow, Nov 04 2015

A212875 Primonacci numbers: composite numbers that appear in the Fibonacci-like sequence generated by their own prime factors.

Original entry on oeis.org

4, 9, 12, 25, 27, 169, 1102, 7921, 22287, 54289, 103823, 777627, 876897, 2550409, 20854593, 34652571, 144237401, 144342653, 167901581, 267911895, 792504416, 821223649, 1103528482, 2040412557, 2852002829, 3493254541, 6033671841, 15658859018, 116085000401
Offset: 1

Views

Author

Herman Beeksma, May 29 2012

Keywords

Comments

Given n, form a sequence that starts with the k prime factors of n in ascending order. After that, each term is the sum of the preceding k terms. If n eventually appears in the sequence, it is a primonacci number. Primes possess this property trivially and are therefore excluded.
Similar to A007629 (repfigit or Keith numbers), but base-independent. If n is in A005478 (Fibonacci primes), then n^2 is a primonacci number.
The only entries that are semiprimes (A001358) are the squares of A005478. - Robert Israel, Mar 08 2016

Examples

			Fibonacci-like sequences for selected values of n:
n=12: 2, 2, 3, 7, 12, ...
n=25: 5, 5, 10, 15, 25, ...
n=1102: 2, 19, 29, 50, 98, 177, 325, 600, 1102, ...

Links

Crossrefs

Cf. A000045, A001358, A005478, A007629.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local a,b,j,k,n,t,v; v:=array(1..h);
for n from 2 to q do if not isprime(n) then b:=ifactors(n)[2]; a:=[];
for k from 1 to nops(b) do for j from 1 to b[k][2] do a:=[op(a),b[k][1]]; od; od; a:=sort([op(a)]);
b:=nops(a);  for k from 1 to b do v[k]:=a[k]; od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]Paolo P. Lava, Mar 08 2016
  • Mathematica
    PrimonacciQ[n_]:=Module[{k,seq},
  seq=FactorInteger[n];
  seq=Map[Table[#[[1]],{#[[2]]}]&, seq];
  seq=Flatten[seq];
  k=Length[seq];
  If[k==1,Return[False]];
  seq=Append[seq,Apply[Plus,seq]];
  While[seq[[-1]]Michael De Vlieger, Mar 08 2016 *)
  • Python
    from sympy import isprime, factorint
from itertools import chain
A212875_list = []
for n in range(2,10**6):
    if not isprime(n):
        x = sorted(chain.from_iterable([p]*e for p,e in factorint(n).items()))
        y = sum(x)
        while y < n:
            x, y = x[1:]+[y], 2*y-x[0]
        if y == n:
            A212875_list.append(n) # Chai Wah Wu, Sep 12 2014

A263880 Safe primes 2p + 1 such that p is a Fibonacci prime.

Original entry on oeis.org

5, 7, 11, 179, 467, 21195998530602981465199287343010006825031720870818843865120019360285948694390966280586508792391539752259819
Offset: 1

Views

Author

Jonathan Sondow, Nov 02 2015

Keywords

Comments

Same as safe primes q whose Sophie Germain prime (2q - 1)/2 is a Fibonacci number.
No other terms up to 2*Fibonacci(2904353) + 1, according to the list of indices of 49 Fibonacci (probable) primes in A001605.
In that range, the only safe Fibonacci prime is 5. Are there larger ones?
There are six primes 2p + 1 such that p is a Fibonacci prime, namely, a(1) through a(6). By contrast, in the same range there are only two primes 2p - 1 such that p is a Fibonacci prime, namely, 2p - 1 = 3 and 5, for p = 2 and 3. Is there some modular restriction to explain this bias in favor of 2p + 1 over 2p - 1 among Fibonacci primes p?

Examples

			179 is in the sequence because it is prime and (179 - 1)/2 = 89 = Fibonacci(11), which is also prime.

Links

Crossrefs

Cf. A000045, A005384, A005385, A005478, A155011.

Programs

  • Mathematica
    2 * Select[Fibonacci[Range[2000]], And @@ PrimeQ[{#, 2 # + 1}] &] + 1

Formula

a(n) = 2*A155011(n) + 1.

A335568 a(n) is the number m such that F(m) is the greatest prime Fibonacci divisor of F(n)^2 + 1 where F(n) is the n-th Fibonacci number, or 0 if no such prime factor exists.

Original entry on oeis.org

3, 3, 5, 5, 7, 7, 5, 7, 11, 11, 13, 13, 11, 13, 17, 17, 5, 17, 17, 7, 23, 23, 7, 23, 23, 5, 29, 29, 3, 29, 29, 11, 7, 11, 11, 7, 13, 13, 0, 13, 43, 43, 5, 43, 47, 47, 7, 47, 47, 17, 7, 17, 17, 11, 3, 11, 11, 3, 3, 0, 7, 7, 13, 13, 7, 13, 23, 23, 0, 23, 23, 0, 5
Offset: 1

Views

Author

Chai Wah Wu, Nov 20 2020

Keywords

Comments

Fibonacci index of the terms in A338762.
All terms are prime or 0. - Alois P. Heinz, Nov 21 2020

Examples

			a(10) = 11 because F(10)^2 + 1 = 55^2 + 1 = 3026 = 2*17*89 and 89 = F(11) is the greatest prime Fibonacci divisor of 3026.

Links

Crossrefs

Cf. A000040, A000045, A005478, A245306, A338762, A338794 (indices of the 0's).

Programs

  • Maple
    a:= proc(n) local i, F, m, t; F, m, t:=
      [1, 2], 0, (<<0|1>, <1|1>>^n)[2, 1]^2+1;
      for i from 3 while F[2]<=t do if isprime(F[2]) and
        irem(t, F[2])=0 then m:=i fi; F:= [F[2], F[1]+F[2]]
      od; m
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 21 2020
  • Mathematica
    a[n_] := Module[{i, F = {1, 2}, m = 0, t}, t = MatrixPower[{{0, 1}, {1, 1}}, n][[2, 1]]^2 + 1; For[i = 3, F[[2]] <= t, i++, If[PrimeQ[F[[2]]] && Mod[t, F[[2]]] == 0, m = i]; F = {F[[2]], F[[1]] + F[[2]]}]; m];
Array[a, 100] (* Jean-François Alcover, Dec 01 2020, after Alois P. Heinz *)

Formula

A000045(a(n)) = A338762(n).

A121533 Fibonacci-Lucas prime twins: Prime Fibonacci numbers Fibonacci(k) such that Lucas numbers Lucas(k) = Fibonacci(k-1) + Fibonacci(k+1) are also prime.

Original entry on oeis.org

3, 5, 13, 89, 233, 1597, 2971215073
Offset: 1

Views

Author

Alexander Adamchuk, Aug 05 2006

Keywords

Comments

Indices for Fibonacci-Lucas prime twins are A080327(n) = {4, 5, 7, 11, 13, 17, 47, ...}. Corresponding Lucas-Fibonacci prime twins are A121534(n) = {7, 11, 29, 199, 521, 3571, 6643838879, ...}. Probable primes Fibonacci(148091) and Lucas(148091) are the next probable Fibonacci-Lucas and Lucas-Fibonacci prime twins. They have 30949 and 30950 digits.

Examples

			a(1) = 3 because Fibonacci(4) = 3 is prime and Lucas(4) = 5 is also prime.

Crossrefs

Cf. A121534, A000045, A005478, A001605, A001606, A000032, A000204, A005479, A080327.

Programs

  • Mathematica
    Do[f=Fibonacci[n]; l=Fibonacci[n-1]+Fibonacci[n+1]; If[PrimeQ[f]&&PrimeQ[l], Print[{f,l}]], {n,10000}]

A139095 Fibonacci numbers whose sum of proper divisors is also a Fibonacci number.

Original entry on oeis.org

1, 1, 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241
Offset: 1

Views

Author

Omar E. Pol, May 11 2008

Keywords

Comments

Fibonacci numbers k such that A001065(k) is a Fibonacci number.
A001065(a(n)) is a Fibonacci number.
Certainly this contains 1 and the terms of A005478. Does it contain any other terms? - R. J. Mathar, Sep 17 2009
The next term, Fibonacci(359) = 4.754...*10^74, is too large to include in the data section. There are no composite Fibonacci numbers below A000045(1423) in this sequence. - Amiram Eldar, Mar 11 2024

Crossrefs

Cf. A000045, A001065, A005478, A074283, A139589.

Programs

  • Maple
    isA000045 := proc(n) local i,f ; for i from 0 do f := combinat[fibonacci](i) ; if f = n then RETURN(true) ; elif f > n then RETURN(false) ; fi ; od; end: A001065 := proc(n) numtheory[sigma](n)-n ; end: isA139095 := proc(n) RETURN( isA000045(n) and isA000045(A001065(n)) ) ; end: for i from 1 to 230 do if isA139095(combinat[fibonacci](i)) then printf("%d,", combinat[fibonacci](i)) ; fi ; od: # R. J. Mathar, May 22 2008
  • Mathematica
    Fsum[n_]:=DivisorSigma[1,n]-n;Select[Fibonacci[Range[140]],IntegerQ[Sqrt[5*Fsum[#]^2 + 4]] || IntegerQ[Sqrt[5*Fsum[#]^2 - 4]]&] (* James C. McMahon, Jun 28 2025 *)

Extensions

More terms from R. J. Mathar, May 22 2008

A165481 Legendre symbol (n,28657).

Original entry on oeis.org

0, 1, 1, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, -1, -1, -1, -1, 1, -1, 1, 1, -1, 1, 1, 1, -1, -1, -1, 1
Offset: 0

Views

Author

Antti Karttunen, Sep 21 2009

Keywords

Comments

28657 is the 8th Fibonacci prime, A005478(8) = A000045(23).

Links

Crossrefs

Partial sums: A165482.

A165586 Legendre symbol (n,514229).

Original entry on oeis.org

0, 1, -1, -1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Antti Karttunen, Sep 22 2009

Keywords

Comments

514229 is the 9th Fibonacci prime, A005478(9) = A000045(29).

Links

Crossrefs

Partial sums: A165587.

Programs

A165596 Jacobi symbol (n,59881).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1
Offset: 0

Views

Author

Antti Karttunen, Sep 22 2009

Keywords

Comments

Semiprime 59881 = 233*257 = A005478(6)*A019434(3) = A117879(11).

Links

Crossrefs

a(n) = A011628(n)*A165573(n). Partial sums: A165597. Cf. A165591, A165471.

A169790 Least number k having n unordered partitions into a nonzero Fibonacci number and a prime.

Original entry on oeis.org

3, 4, 10, 24, 74, 444, 1614, 15684, 29400, 50124, 259224, 5332128, 11110428, 50395440, 451174728, 1296895890
Offset: 1

Views

Author

R. J. Mathar and Jon E. Schoenfield, May 14 2010

Keywords

Comments

Variant of A168382.
Fibonacci(1) + prime(4) = Fibonacci(2) + prime(4) = Fibonacci(4) + prime(3) = Fibonacci(5) + prime(2) = 8 are two "distinct" representations of k=8, because Fibonacci(1) = Fibonacci(2) = 1 is treated as indistinguishable, and Fibonacci(4) = prime(2) = 3 are also indistinguishable: k = 1+7 = 3+5.
This matters because of the existence of Fibonacci primes (see A005478).
a(17) > 10^10. [Donovan Johnson, May 17 2010]

Examples

			1+443 = 5+439 = 13+431 = 55+389 = 233+211 = 377+67 are n=6 distinct representations of 444.

Crossrefs

Cf. A168382, A169791.

Extensions

a(8)-a(14) from Max Alekseyev, May 15 2010
a(15)-a(16) from Donovan Johnson, May 17 2010
Prime index in the comment corrected by R. J. Mathar, Jun 02 2010
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