A384612 a(n) is the smallest integer k such that k^n is an abelian square; or -1 if no such k exists.

11, 836, 11, 207, 624, 818222, 1001, 2776, 100001, 32323107, 100001, 85692627, 10000001, 501249084, 10000001, 27962757, 41695607, 70983559, 72768046, 977688137, 219873071, 112562383, 2338280974, 2435385853, 1231380445, 4557057314, 361499019, 8096434047, 5278552513

Offset: 1

Author

Gonzalo Martínez, Jun 04 2025

Keywords

• nonn
• base

Comments

Terms are the base of the smallest n-th power whose string of decimal digits is an abelian square; i.e., of the form m concatenated with a permutation of m (A272655).

If n is odd and A001700((n-1)/2) has d digits, then 0 < k <= 10^(2*d-1) + 1. - Robert Israel, Jun 05 2025

a(23) >= 1.83 * 10^9. Using Robert Israel's comment above a(23) <= 10^13 + 1. - David A. Corneth, Jun 06 2025

Examples

			a(1) = 11, since 11^1  = 1|1
a(2) = 836, since 836^2 = 698|896
a(3) = 11, since 11^3 = 13|31
a(4) = 207, since 207^4 = 18360|36801
a(5) = 624, since 624^5 = 9460692|9690624
a(6) = 818222, since 818222^6 = 300072996174564185|100579862765194304
a(7) = 1001, since 1001^7 = 10070210350|35021007001
a(8) = 2776, since 2776^8 = 35265958674713|24535718936576
a(9) = 100001, since 100001^9 = 10000900036000840012600|12600084000360000900001.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..34

Crossrefs

Cf. A001597, A001700, A075786, A272655, A342942.

Programs

Maple

f:= proc(n) local k,d,x,L;
      k:= 2;
      do
        x:= k^n;
        d:= ilog10(x)+1;
        if d::odd then k:= ceil(10^(d/n)); next fi;
        L:= convert(x,base,10);
        if sort(L[1..d/2]) = sort(L[d/2+1..d]) then return k fi;
        k:= k+1
      od;
end proc:
map(f, [$1..30]); # Robert Israel, Jun 05 2025

Python

from itertools import count
def ok(k, n):
    s = str(k**n)
    if len(s) % 2 != 0:
        return False
    mid = len(s) // 2
    return sorted(s[:mid]) == sorted(s[mid:])
def a(n):
    return next(k for k in count(2) if ok(k, n))
print([a(n) for n in range(1, 10)])

Extensions

a(20)-a(22) from David A. Corneth, Jun 06 2025

a(23) from Gonzalo Martínez, Jun 06 2025

a(24)-a(29) from Jinyuan Wang, Jun 14 2025