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%I #21 May 03 2026 23:18:58
%S 46656,2176782336,10000000000,8916100448256,101559956668416,
%T 11112006825558016,437893890380859375,4738381338321616896,
%U 100000000000000000000,39346408075296537575424,221073919720733357899776,79496847203390844133441536,104857600000000000000000000,5842587018385982521381124421
%N Powers k^(m*k), with k in A024619 and m >= 1.
%C Proper subset of A131605.
%C 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.
%C 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.
%C 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.
%C See A368107 for proper prime powers p^(m*p), m >= 1.
%H Michael De Vlieger, <a href="/A394512/b394512.txt">Table of n, a(n) for n = 1..1557</a>
%e Table of n, a(n) for select n:
%e n a(n)
%e -----------------------------------------------------------
%e 1 46656 = 6^6 = 2^6 * 3^6
%e 2 2176782336 = 6^(2*6) = 2^12 * 3^12
%e 3 10000000000 = 10^10 = 2^10 * 5^10
%e 4 8916100448256 = 12^12 = 2^24 * 3^12
%e 5 101559956668416 = 6^(3*6) = 2^18 * 3^18
%e 6 11112006825558016 = 14^14 = 2^14 * 7^14
%e 7 437893890380859375 = 15^15 = 3^15 * 5^15
%e 8 4738381338321616896 = 6^(4*6) = 2^24 * 3^24
%e 9 = 10^(2*10) = 2^20 * 5^20
%e 28 = 30^30 = 2^30 * 3^30 * 5^30
%e 41 36^36 = 6^(12*6) = 2^72 * 3^72
%e 131 = 72^72 = 2^216 * 3^144
%t 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]]
%Y Cf. A001597, A024619, A052486, A120944, A131605, A303606, A332785, A368107, A383394, A386762.
%K nonn,easy
%O 1,1
%A _Michael De Vlieger_, Apr 28 2026