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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.

%I A373049 #39 May 28 2025 00:59:26
%S A373049 0,1,2,6,10,12,19,21,22,27,31,46,49,50,73,79,85,102,108,116,117,160,
%T A373049 161,179,181,237,247,250,257,281,285,302,309,351,354,359,373,376,377,
%U A373049 380,415,419,434,449,470,479,497,498,515,521,543,565,569,584,590,599,602,609,615,665,696
%N A373049 Integers k such that the product of the nonzero digits of the k-th Fibonacci number (A000045) is a perfect power.
%C A373049 For most of the terms in this list, the product of their nonzero digits is a perfect square.
%C A373049 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.
%H A373049 Robert Israel, <a href="/A373049/b373049.txt">Table of n, a(n) for n = 1..10000</a>
%e A373049 21 is a term, because Fibonacci(21) = 10946 and the product of its nonzero digits is 1*9*4*6 = 6^3.
%e A373049 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.
%p A373049 filter:= proc(n) local L,t,q2,q3,q5,q7;
%p A373049   L:=convert(combinat:-fibonacci(n),base,10);
%p A373049   q2:= 0: q3:= 0: q5:= 0: q7:= 0:
%p A373049   for t in L do
%p A373049     if t = 2 then q2:= q2+1
%p A373049     elif t = 3 then q3:= q3+1
%p A373049     elif t = 4 then q2:= q2+2
%p A373049     elif t = 5 then q5:= q5+1
%p A373049     elif t = 6 then q2:= q2+1; q3:= q3+1
%p A373049     elif t = 7 then q7:= q7+1
%p A373049     elif t = 8 then q2:= q2+3
%p A373049     elif t = 9 then q3:= q3+2
%p A373049     fi
%p A373049   od;
%p A373049   igcd(q2,q3,q5,q7) > 1
%p A373049 end proc:
%p A373049 filter(0):= true: filter(1):= true: filter(2):= true:
%p A373049 select(filter, [$0..1000]); # _Robert Israel_, May 26 2025
%t A373049 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 *)
%o A373049 (PARI) isok(k) = my(x=vecprod(select(x->(x>0), digits(fibonacci(k))))); (x==1) || ispower(x); \\ _Michel Marcus_, May 20 2024
%Y A373049 Cf. A000045, A246558, A076564, A370071, A373116.
%K A373049 nonn,base
%O A373049 1,3
%A A373049 _Gonzalo Martínez_, May 20 2024
%E A373049 More terms from _Michel Marcus_, May 20 2024