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

1,3

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.

Links

Table of n, a(n) for n=1..61.

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A263309 A253913 A190789 * A071638 A028348 A214963

Adjacent sequences: A373046 A373047 A373048 * A373050 A373051 A373052

Keyword

nonn,base

Author

Gonzalo Martínez, May 20 2024

Extensions

More terms from Michel Marcus, May 20 2024

Status

approved