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.
Triangle begins:
[ 1 ],
[ 1 ],
[ 1, 2 ],
[ 1, 3 ],
[ 1, 5 ],
[ 1, 2, 4, 8 ],
[ 1, 13 ],
[ 1, 3, 7, 21 ],
[ 1, 2, 17, 34 ],
[ 1, 5, 11, 55 ],
[ 1, 89 ],
/* As triangle */ [Divisors(Fibonacci(n)): n in [1..30]]; // Vincenzo Librandi, Nov 15 2024
Flatten[Table[Divisors[Fibonacci[n]], {n, 20}]] (* T. D. Noe, Mar 14 2014 *)
From _Michael De Vlieger_, May 18 2017: (Start)
a(6) = 8 because Fibonacci(6) = 8, the multiplicity of the prime factor of 8 is 3; the smallest p^3 = 2^3 = 8.
a(7) = 2 because Fibonacci(7) = 13, the multiplicity of the prime factor of 13 is 1; the smallest p^1 = 2^1 = 2.
a(15) = 30 because Fibonacci(15) = 610. The multiplicities of the prime factors of 610, in order from greatest to least, are {1, 1, 1}, the smallest prime power product p^1 * q^1 * r^1 = 2 * 3 * 5 = 30.
a(18) = 120 because Fibonacci(18) = 2584 = 2^3 * 17 * 19 -> 2^3 * 3 * 5 = 120. (End)
Table[If[# == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &,
Sort[FactorInteger[#][[All, -1]], Greater]]] &@ Fibonacci@ n, {n, 79}] (* Michael De Vlieger, May 18 2017 *)
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011 f0 = 0; f1 = 1; for(n=1, 10000, write("b278245.txt", n, " ", A046523(f1)); old_f0 = f0; f0 = f1; f1 = f1 + old_f0; );
(define (A278245 n) (A046523 (A000045 n)))
n=12, A000045(12)=144: 5 of the 15 divisors of 144 are also Fibonacci numbers, a(12) = #{1, 2, 3, 8, 144} = 5.
with(combinat, fibonacci):a[1] := 1:for i from 2 to 229 do s := 0:for j from 2 to i do if((fibonacci(i) mod fibonacci(j))=0) then s := s+1:fi:od:a[i] := s:od:seq(a[l],l=2..229);
Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 100}] (Noe)
{a(n)=if(n<1, 0, numdiv(n)+n%2-1)} /* Michael Somos, Sep 03 2006 */
{a(n)=if(n<1, 0, sumdiv(n,d, d!=2))} /* Michael Somos, Sep 03 2006 */
A000045 := proc(n) option remember ; coeftayl( x/(1-x-x^2),x=0,n) ; end: isA000045 := proc(n) local a; for a from 0 do if A000045(a) > n then RETURN(false) ; elif A000045(a)=n then RETURN(true) ; fi ; od: end: A000005 := proc(n) numtheory[tau](n) ; end: isA139589 := proc(n) RETURN(isA000045(n) and isA000045(A000005(n))) ; end: for i from 1 to 130 do a000045 := A000045(i) ; if isA139589(a000045) then printf("%d,",a000045) ; fi ; od: # R. J. Mathar, May 11 2008 with(combinat): with(numtheory): F:={seq(fibonacci(k),k=1..100)}: a:=proc(n) if member(tau(fibonacci(n)),F)=true then fibonacci(n) else end if end proc: seq(a(n),n=1..70); # Emeric Deutsch, May 12 2008
With[{s = Array[Fibonacci, 80]}, Select[s, ! FreeQ[s, DivisorSigma[0, #]] &]] (* Michael De Vlieger, Oct 15 2019 *)
a(8)=8 because the 8th tribonacci number 24 has divisors {1, 2, 3, 4, 6, 8, 12, 24}.
DivisorSigma[0, LinearRecurrence[{1, 1, 1}, {1, 1, 2}, 70]] (* Amiram Eldar, Oct 23 2023 *)
34 belongs to the sequence because the number of its divisors, 4, is not a Fibonacci number.
A000045 := proc(n) option remember ; coeftayl( x/(1-x-x^2),x=0,n) ; end: isA000045 := proc(n) local a; for a from 0 do if A000045(a) > n then RETURN(false) ; elif A000045(a)=n then RETURN(true) ; fi ; od: end: A000005 := proc(n) numtheory[tau](n) ; end: isA139590 := proc(n) RETURN(isA000045(n) and not isA000045(A000005(n))) ; end: for i from 1 to 130 do a000045 := A000045(i) ; if isA139590(a000045) then printf("%d,",a000045) ; fi ; od: # R. J. Mathar, May 11 2008 with(combinat): with(numtheory): F:={seq(fibonacci(j),j=1..30)}: a:= proc(n) if member(tau(fibonacci(n)),F) = false then fibonacci(n) else end if end proc: seq(a(n),n=1..50); # Emeric Deutsch
With[{fibs=Fibonacci[Range[60]]},Transpose[Select[Thread[{fibs, DivisorSigma[ 0,fibs]}], !MemberQ[ fibs,#[[2]]]&]][[1]]] (* Harvey P. Dale, Aug 04 2013 *)
144 is in the sequence because it is a Fibonacci number with 15 divisors and all smaller Fibonacci numbers have fewer divisors. - _Emeric Deutsch_, Jan 20 2009
with(numtheory): with(combinat): a := proc (n) if n = 1 then 1 else if max(seq(tau(fibonacci(j)), j = 1 .. n-1)) < tau(fibonacci(n)) then fibonacci(n) else end if end if end proc: seq(a(n), n = 1 .. 170); # Emeric Deutsch, Jan 20 2009
DeleteDuplicates[{#,DivisorSigma[0,#]}&/@Fibonacci[Range[200]],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Aug 10 2025 *)
with(combinat):with(numtheory): A160683 := proc(n) option remember: local k: if(n=1)then return 1:fi: for k from procname(n-1)+1 do if(fibonacci(k) mod tau(fibonacci(k))=0)then return k:fi: od: end: seq(A160683(n), n=1..6); # Nathaniel Johnston, May 09 2011
Select[Range@ 120, IntegerQ[#/DivisorSigma[0, #]] &@ Fibonacci@ # &]
isok(n) = my(f=fibonacci(n)); f % numdiv(f) == 0; \\ Michel Marcus, Jul 31 2015
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