A387882 a(n) is the smallest k > 0 such that exactly n distinct permutations of the digits of k, all having the same number of digits as k, are perfect squares (no leading zeros allowed).

1, 144, 169, 10044, 13468, 10069, 1003468, 1000069, 10002456, 11234689, 10234467, 100012669, 100012444, 102344899, 102245679, 100124469, 102234789, 100345689, 102344569, 100024569, 1001223459, 102345678, 1001234569, 1000123569, 1002344679, 1001234689

Offset

1,2

Comments

a(2n) <= 10^(2n) + 44. Proof. Consider the number m = 12*10^(n - 1). There are a total of 2*n permutations of m that have the same number of digits as m, where each of them satisfies that its square is composed of (2n + 1) digits, which are: 1, 4, 4, and (2n - 2)'s zeros. In this way, we have generated 2*n perfect squares that share all their digits.

Links

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

Example

n|    k    | [permutations of k that are perfect squares]
1|    1    | [1]
2|   144   | [144, 441]
3|   169   | [169, 196, 961]
4|  10044  | [10404, 14400, 40401, 44100]
5|  13468  | [16384, 31684, 36481, 38416, 43681]
6|  10069  | [10609, 16900, 19600, 61009, 90601, 96100]
7| 1003468 | [1638400, 3168400, 3648100, 3806401, 3841600, 4036081, 4368100]
8| 1000069 | [1006009, 1060900, 1690000, 1960000, 6100900, 9006001, 9060100, 9610000]

Prog

(PARI) upto(n) = {my(d=digits(n), qd=#d, maxn=10^qd-1, maxi=sqrtint(maxn), m=Map(), res=[] ); for(i=1, maxi, c=i^2; s=A328447(c); if(mapisdefined(m, s), q=mapget(m, s); mapput(m, s, q+1); , mapput(m, s, 1))); foreach(m, item, if(item[1][2]>#res, res=concat(res, vector(item[1][2]-#res, i, oo)); ); res[item[1][2]]=min(res[item[1][2]], item[1][1]); ); res}
A328447(n)={if(n=vecsort(digits(n)), n[1]|| for(k=2, #n, n[k]&&[n[1]=n[k], n[k]=0, break])); fromdigits(n)} \\ (A328447 from M. F. Hasler, Oct 15 2019)
\\ David A. Corneth, Dec 09 2025
(Python)
from math import isqrt
from collections import Counter
from itertools import count, islice
def b(s): # A328447 for string input
    t = "".join(sorted(s)).strip("0")
    return int(t[0]+"0"*s.count("0")+t[1:])
def agen():
    adict, n, pcdict, digs = dict(), 1, Counter(), -1
    for d in count(1):
        pcdict = Counter()
        for m in range(isqrt(10**(d-1)-1)+1, isqrt(10**d-1)+1):
            pcdict[b(str(m*m))] += 1
        if pcdict:
            vals = set(pcdict.values())
            for i in range(n, max(vals)+1):
                if i in vals:
                    adict[i] = next(k for k in sorted(pcdict) if pcdict[k] == i)
                    while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 30))) # Michael S. Branicky, Dec 09 2025

Crossrefs

Cf. A000290, A128993, A179239, A328447.

Sequence in context: A035090 A064021 A156316 * A386302 A323614 A245214

Adjacent sequences: A387879 A387880 A387881 * A387883 A387884 A387885

Keyword

nonn,base

Author

Gonzalo Martínez, Dec 08 2025

Extensions

a(7), a(9), a(14)-a(18), a(23), a(26) corrected by Michael S. Branicky, Dec 08 2025

Status

approved