A370071 -id:A370071 - cp's OEIS frontend

A371588 Smallest Fibonacci number > 1 such that some permutation of its digits is a perfect n-th power.

2, 144, 8, 610, 5358359254990966640871840, 68330027629092351019822533679447, 15156039800290547036315704478931467953361427680642, 23770696554372451866815101694984845480039225387896643963981, 119447720249892581203851665820676436622934188700177088360

Offset: 1

Chai Wah Wu, Mar 28 2024

Subsequence of A370071 after reordering (as the sequence is not monotonic; e.g., a(2) > a(3) and a(8) > a(9)). Leading 0 digits are allowed in the perfect power. For example, a(4) = 610 since 016 = 2^4. (If leading 0 digits were not allowed, a(4) would be 160500643816367088.)

			a(1) = 2 since 2 = 2^1.
a(2) = 144 since 144 = 12^2.
a(3) = 8 since 8 = 2^3.
a(4) = 610 since 016 = 2^4.
a(5) = 5358359254990966640871840 since 0735948608251696955804943 = 59343^5
a(6) = 68330027629092351019822533679447 since 00059398947526192142327360782336 = 62464^6.

Chai Wah Wu, Table of n, a(n) for n = 1..16

Cf. A000045, A227875, A001597, A118715, A370071.

from itertools import count
from sympy import integer_nthroot
def A371588(n):
    a, b = 1, 2
    while True:
        s = sorted(str(b))
        l = len(s)
        m = int(''.join(s[::-1]))
        u = int(''.join(s))
        for i in count(max(2,integer_nthroot(u,n)[0])):
            if (k:=i**n) > m:
                break
            t = sorted(str(k))
            if ['0']*(l-len(t))+t == s:
                return b
                break
        a, b = b, a+b

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

A371589 Smallest number m > 1 such that some permutation of the digits of m^n is a Fibonacci number.

Original entry on oeis.org

2, 12, 2, 19002, 433153, 472133, 10064513, 61054259, 67878079, 8152101, 46077414, 11395185, 28556455, 11730986, 179311318, 1542839498, 443163383, 2426412518, 433059953, 443302473, 2654438078, 2764480203, 5945916934

Offset: 1

Chai Wah Wu, Mar 28 2024

Unlike in A370071 or A371588, no leading 0's are allowed in m^n or the Fibonacci number.

			a(4) = 19002 since 19002^4 = 130375880664608016 and a permutation of its digits results in 160500643816367088, a Fibonacci number.

Cf. A000045, A227875, A001597, A118715, A370071, A371588.

from itertools import count
from sympy import integer_nthroot
def A371589(n):
    a, b = 1, 2
    while True:
        s = sorted(str(b))
        m = int(''.join(s[::-1]))
        u = int(''.join(s))
        for i in count(max(2,integer_nthroot(u,n)[0])):
            if (k:=i**n) > m:
                break
            t = sorted(str(k))
            if t == s:
                return i
                break
        a, b = b, a+b

a(15)-a(23) from Bert Dobbelaere, Apr 10 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 *)

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A371588 Smallest Fibonacci number > 1 such that some permutation of its digits is a perfect n-th power.

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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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A371589 Smallest number m > 1 such that some permutation of the digits of m^n is a Fibonacci number.

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

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