A373049 Integers k such that the product of the nonzero digits of the k-th Fibonacci number (A000045) is a perfect power.

0, 1, 2, 6, 10, 12, 19, 21, 22, 27, 31, 46, 49, 50, 73, 79, 85, 102, 108, 116, 117, 160, 161, 179, 181, 237, 247, 250, 257, 281, 285, 302, 309, 351, 354, 359, 373, 376, 377, 380, 415, 419, 434, 449, 470, 479, 497, 498, 515, 521, 543, 565, 569, 584, 590, 599, 602, 609, 615, 665, 696

Offset: 1

Author

Gonzalo Martínez, May 20 2024

Keywords

• nonn
• base

Comments

For most of the terms in this list, the product of their nonzero digits is a perfect square.

Conjecture: this sequence has infinitely many terms. Since the product of the nonzero digits of Fibonacci(k) is of the form 2^a * 3^b * 5^c * 7^d, a sufficient condition for Fibonacci(k) to belong to the sequence is that a, b, c and d are all even.

Examples

			21 is a term, because Fibonacci(21) = 10946 and the product of its nonzero digits is 1*9*4*6 = 6^3.
46 is a term, because Fibonacci(46) = 1836311903 and the product of its nonzero digits is 1*8*3*6*3*1*1*9*3 = 108^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000045, A246558, A076564, A370071, A373116.

Programs

Maple

filter:= proc(n) local L,t,q2,q3,q5,q7;
  L:=convert(combinat:-fibonacci(n),base,10);
  q2:= 0: q3:= 0: q5:= 0: q7:= 0:
  for t in L do
    if t = 2 then q2:= q2+1
    elif t = 3 then q3:= q3+1
    elif t = 4 then q2:= q2+2
    elif t = 5 then q5:= q5+1
    elif t = 6 then q2:= q2+1; q3:= q3+1
    elif t = 7 then q7:= q7+1
    elif t = 8 then q2:= q2+3
    elif t = 9 then q3:= q3+2
    fi
  od;
  igcd(q2,q3,q5,q7) > 1
end proc:
filter(0):= true: filter(1):= true: filter(2):= true:
select(filter, [$0..1000]); # Robert Israel, May 26 2025

Mathematica

powQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[0, 700], powQ[Times @@ Select[IntegerDigits[Fibonacci[#]], #1 > 0 &]] &] (* Amiram Eldar, May 25 2024 *)

PARI

isok(k) = my(x=vecprod(select(x->(x>0), digits(fibonacci(k))))); (x==1) || ispower(x); \\ Michel Marcus, May 20 2024

Extensions

More terms from Michel Marcus, May 20 2024