A076564 -id:A076564 - cp's OEIS frontend

A177194 Fibonacci numbers whose decimal expansion does not contain any digit 0.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 987, 1597, 2584, 4181, 6765, 17711, 28657, 46368, 121393, 196418, 317811, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 267914296, 433494437, 53316291173, 86267571272

Offset: 1

Carmine Suriano, May 04 2010

The probability that fib(n) contains no 0's decreases to zero as n goes to infinity. Its maximum value seems to be F(184) having 39 digits, including no zeros.

			a(7)=13 since fib(7) does not contain the digit 0.

Cf. A176253, A000045

a(n) = A000045(A076564(n)). [From R. J. Mathar, Oct 18 2010]

A246558 Product of the digits of the n-th Fibonacci number.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 3, 2, 12, 25, 72, 16, 18, 147, 0, 504, 315, 320, 32, 1260, 0, 49, 3360, 3456, 0, 162, 1728, 168, 720, 0, 7776, 0, 33600, 0, 30240, 0, 15680, 0, 311040, 0, 0, 326592, 435456, 0, 0, 0, 0, 0, 0, 0, 0, 0, 102060, 3951360, 24883200, 1411200

Offset: 0

Indrani Das, Nov 12 2014

a(n) > 0 iff n in A076564.

Probably, the last nonzero term is a(184). - Giovanni Resta, Jul 14 2015

			Fibonacci(7) = 13, thus a(7) = 1*3 = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000045, A004090, A076564.
Cf. A007954, A259773.
Cf. A261598.

a246558 = a007954 . a000045 -- Reinhard Zumkeller, Nov 17 2014

[0] cat [&*Intseq(Fibonacci(n)): n in [1..100]]; // Vincenzo Librandi, Jan 04 2020

Array[Times@@IntegerDigits@Fibonacci[#]&, 100, 0] (* Vincenzo Librandi, Jan 04 2020 *)

a(n) = if (n, vecprod(digits(fibonacci(n))), 0); \\ Michel Marcus, Feb 11 2025

a(n) = A007954(A000045(n)). - Reinhard Zumkeller, Nov 17 2014

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

Original entry on oeis.org

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

Gonzalo Martínez, May 20 2024

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.

			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.

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

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

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

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 *)

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

More terms from Michel Marcus, May 20 2024

A373116 Fibonacci numbers whose digits product is a positive perfect power (A001597).

Original entry on oeis.org

1, 8, 55, 144, 4181, 17711, 196418, 1346269, 259695496911122585

Offset: 1

Gonzalo Martínez, May 25 2024

Since the product of the digits of Fibonacci(k) is required to be positive, Fibonacci(k) does not have zero as a digit. For this reason this list is probably finite, since it is conjectured that there are only finitely many Fibonacci numbers that do not contain the digit zero (see A076564). If the conjecture is true, the largest number possessing the property would be Fibonacci(85) = 259695496911122585 whose digit product is 194400^2.

Unlike A373049, here the product uses all the digits of Fibonacci(k).

			196418 is a term, because Fibonacci(27) = 196418 and the product of its digits is 1*9*6*4*1*8 = 12^3.

Cf. A000045, A001597, A076564, A246558, A370071, A373049.

powQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; Select[Fibonacci[Range[2, 100]], powQ[Times @@ IntegerDigits[#]] &] (* Amiram Eldar, May 25 2024 *)

A130631 Multiplicative persistence of Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 3, 2, 2, 4, 1, 2, 3, 2, 2, 2, 1, 4, 2, 3, 1, 3, 3, 4, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jun 19 2007

From the 184th terms on all the Fibonacci numbers have some digits equal to zero (see A076564), thus their persistence is equal to 1.

			3524578 -> 3*5*2*4*5*7*8 = 33600 -> 3*3*6*0*0 = 0 -> persistence = 2.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000045, A076564.

P:=proc(n)local f0,f1,f2,i,k,w,ok,cont; f0:=0; f1:=1; print(0); print(0); for i from 0 by 1 to n do f2:=f1+f0; f0:=f1; f1:=f2; w:=1; ok:=1; k:=f2; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(100);

Table[Length[NestWhileList[Times@@IntegerDigits[#]&, Fibonacci[n], #>=10&]], {n, 0, 102}]-1 (* James C. McMahon, Feb 11 2025 *)

a(n) = A031346(A000045(n)). - Michel Marcus, Feb 11 2025

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A177194 Fibonacci numbers whose decimal expansion does not contain any digit 0.

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A246558 Product of the digits of the n-th Fibonacci number.

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

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A373116 Fibonacci numbers whose digits product is a positive perfect power (A001597).

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Mathematica

A130631 Multiplicative persistence of Fibonacci numbers.

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Maple

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