OFFSET
1,1
COMMENTS
Proper subset of A131605.
Smallest term with k that is squarefree and composite (and thus k in A120944 and k^m in A303606) is a(1) = 6^6 = 46656.
Smallest term with k that is neither squarefree nor powerful (thus k in A332785 and k^m in A386762) is a(4) = 12^12 = 8916100448256.
Smallest term with Achilles k (and thus k in A052486 and k^m in A383394) is a(131) = 72^72, a number with 134 decimal digits.
See A368107 for proper prime powers p^(m*p), m >= 1.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1557
EXAMPLE
Table of n, a(n) for select n:
n a(n)
-----------------------------------------------------------
1 46656 = 6^6 = 2^6 * 3^6
2 2176782336 = 6^(2*6) = 2^12 * 3^12
3 10000000000 = 10^10 = 2^10 * 5^10
4 8916100448256 = 12^12 = 2^24 * 3^12
5 101559956668416 = 6^(3*6) = 2^18 * 3^18
6 11112006825558016 = 14^14 = 2^14 * 7^14
7 437893890380859375 = 15^15 = 3^15 * 5^15
8 4738381338321616896 = 6^(4*6) = 2^24 * 3^24
9 = 10^(2*10) = 2^20 * 5^20
28 = 30^30 = 2^30 * 3^30 * 5^30
41 36^36 = 6^(12*6) = 2^72 * 3^72
131 = 72^72 = 2^216 * 3^144
MATHEMATICA
nn = 2^100; i = 6; Union@ Reap[While[j = 1; While[Set[k, i^(j*i)] <= nn, Sow[k]; j++]; j > 1, i++; If[PrimePowerQ[i], While[PrimePowerQ[i], i++] ] ] ][[-1, 1]]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Apr 28 2026
STATUS
approved