A351327 Numbers whose trajectory under iteration of the product of squares of nonzero digits map includes 1.

1, 5, 10, 11, 15, 25, 50, 51, 52, 100, 101, 105, 110, 111, 115, 125, 150, 151, 152, 205, 215, 250, 251, 255, 357, 375, 455, 500, 501, 502, 510, 511, 512, 520, 521, 525, 537, 545, 552, 554, 573, 735, 753, 1000, 1001, 1005, 1010, 1011, 1015, 1025, 1050, 1051

Offset: 1

Author

Luca Onnis, Feb 07 2022

Keywords

• nonn
• base

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)

Examples

			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.
		

Links

• Luca Onnis, On a variant of the happy numbers and their generalizations, arXiv:2203.03381 [math.GM], 2022.

Crossrefs

Cf. A007770, A051801, A115385.

Programs

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 &]

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

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