A386395 Smallest final number reachable from n by successively replacing a substring of digits with its square root.

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 12, 13, 12, 15, 2, 17, 18, 13, 20, 21, 22, 23, 22, 5, 26, 27, 28, 23, 30, 31, 32, 33, 32, 35, 6, 37, 38, 33, 20, 21, 22, 23, 22, 5, 26, 27, 28, 7, 50, 51, 52, 53, 52, 55, 56, 57, 58, 53, 60, 61, 62, 63, 8, 65, 66, 67, 68, 63, 70, 71, 72, 73, 72, 75, 76, 77, 78, 73, 80, 3, 82, 83, 82

Offset

1,2

Comments

The substring replaced must be a square and cannot begin with a 0 digit.

There may be multiple different replacements possible and the aim is to find whatever steps reach the smallest final value.

The sequence could also start with a(0) = 0. - M. F. Hasler, Sep 23 2025

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Michael S. Branicky, Scatter plot of n, a(n) for n = 1..10^6.

Example

425 -> 25 -> 5.
1004 -> 102.
6251 -> 251 -> 51.
9616 -> 3616 -> 196 -> 136 -> 16 -> 4 -> 2.
9996 -> 9936 -> 996 -> 936 -> 96 -> 36 -> 6.

Prog

(Python)
from functools import cache
from math import isqrt
def children(n):
    s = str(n)
    return set(int(s[:i]+str(r)+s[j:]) for i in range(len(s)) for j in range(i+1, len(s)+1) if (w:=s[i:j])[0]!='0' and (r:=isqrt(t:=int(w)))**2 == t)
@cache
def a(n): return min((a(c) for c in children(n)-{n}), default=n)
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jul 20 2025, edited Sep 28 2025
(PARI) apply( {A386395(n, m=n)=if(n<14, issquare(n, &m), my(d=digits(n)); for(i=1, #d, d[i] && for(j=i, #d, issquare(fromdigits(d[i..j]), &n)&& n>1&& m=min(A386395(fromdigits(concat([d[1..i-1], digits(n), d[j+1..-1]]))), m)))); m}, [0..99]) \\ M. F. Hasler, Sep 23 2025

Crossrefs

Cf. A000290, A048385.

Sequence in context: A372379 A350390 A366765 * A361253 A097448 A066729

Adjacent sequences: A386392 A386393 A386394 * A386396 A386397 A386398

Keyword

nonn,base

Author

Ali Sada, Jul 20 2025

Status

approved