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
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
Programs
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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 *)
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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
Comments