OFFSET
1,2
COMMENTS
To determine whether a given number k is a term of this sequence, start with k, take the square of the product of its nonzero digits, apply the same process to the result, and continue until 1 is reached or a loop is entered. If 1 is reached, k is a term of this sequence.
Every power 10^k is a term of this sequence.
If k is a term, the numbers obtained by inserting zeros anywhere in k are terms.
If k is a term, the numbers obtained by inserting ones anywhere in k are terms.
If k is a term, each distinct permutation of the digits of k gives another term.
If k is a term, the number of iterations required to converge to 1 is less than or equal to 3 (conjectured).
From Michael S. Branicky, Feb 07 2022: (Start)
The product of squares of nonzero digits map, f, has fixed points given in A115385.
The map f has (at least) the following cycles:
- 324, 576, 44100, 256, 3600;
- 11664, 20736, 63504, 129600;
- 15876, 2822400, 65536, 7290000;
- 5308416, 8294400;
- 49787136000000, 64524128256, 849346560000, 386983526400, 55725627801600.
(End)
LINKS
Luca Onnis, On a variant of the happy numbers and their generalizations, arXiv:2203.03381 [math.GM], 2022.
EXAMPLE
255 is a term of the sequence: the square of the product of its nonzero digits is (2*5*5)^2=2500, the square of the product of its nonzero digits is (2*5)^2=100, and the square of the product of its nonzero digits is 1^2=1.
2 is not a term of the sequence because its trajectory under the map is 2 -> 4 -> 16 -> 36 -> 324 -> 576 -> 44100 -> 256 -> 3600 -> 324 (reached earlier), so it enters a loop and never reaches 1.
MAPLE
b:= proc() false end:
q:= proc(n) local m, s; m, s:= n, {};
do if m=1 then return true
elif m in s or b(m) then b(n):= true; return false
else s, m:= {s[], m}, mul(max(1, i)^2, i=convert(m, base, 10))
fi
od
end:
select(q, [$1..2000])[]; # Alois P. Heinz, Feb 11 2022
MATHEMATICA
Select[Range[1000],
FixedPoint[
Product[ReplaceAll[0 -> 1][IntegerDigits[#]][[i]]^2, {i, 1,
Length[ReplaceAll[0 -> 1][IntegerDigits[#]]]}] &, #, 10] == 1 &]
PROG
(Python)
from math import prod
def psd(n): return prod(int(d)**2 for d in str(n) if d != "0")
def ok(n):
seen = set()
while n not in seen: # iterate until fixed point or in cycle
seen.add(n)
n = psd(n)
return n == 1
def aupto(n): return [k for k in range(1, n+1) if ok(k)]
print(aupto(1205)) # Michael S. Branicky, Feb 07 2022
(PARI) f(n) = vecprod(apply(d -> if (d, d^2, 1), digits(n)))
is(n) = { my (m=f(n)); while (1, if (n==1, return (1), n==m, return (0), n=f(n); m=f(f(m)))) } \\ Rémy Sigrist, Feb 11 2022
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Luca Onnis, Feb 07 2022
STATUS
approved