

A351327


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


2



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
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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

Table of n, a(n) for n=1..52.
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

Cf. A007770, A051801, A115385.
Sequence in context: A275200 A225838 A036788 * A136811 A130228 A136826
Adjacent sequences: A351324 A351325 A351326 * A351328 A351329 A351330


KEYWORD

nonn,base,changed


AUTHOR

Luca Onnis, Feb 07 2022


STATUS

approved



