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%I #19 Nov 11 2025 21:29:27
%S 8,16,27,32,64,81,125,128,216,243,256,343,512,625,729,1000,1024,1296,
%T 1331,2048,2187,2197,2401,2744,3125,3375,4096,4913,6561,6859,7776,
%U 8192,9261,10000,10648,12167,14641,15625,16384,16807,17576,19683,24389,27000,28561,29791
%N Cubefull powers of squarefree numbers.
%C Intersection of A036966 and A072777.
%C A072777, the sequence squares of squarefree numbers that are not squarefree, is the union of this sequence, and A062503.
%C Union of A246549 (i.e., cubefull prime powers) and A388304 (cubefull numbers in A303606).
%H Michael De Vlieger, <a href="/A390540/b390540.txt">Table of n, a(n) for n = 1..10000</a>
%F Sum_{n>=1} 1/a(n) = A368250 - A082020 + 1 = 0.32881611332981706102... . - _Amiram Eldar_, Nov 11 2025
%e Table of n, a(n) for select n:
%e n a(n)
%e ------------------------------------
%e 1 8 = 2^3
%e 2 16 = 2^4
%e 3 27 = 3^3
%e 4 32 = 2^5
%e 5 64 = 2^6
%e 6 81 = 3^4
%e 7 125 = 5^3
%e 8 128 = 2^7
%e 9 216 = 6^3 = 2^3 * 3^3
%e 16 1000 = 10^3 = 2^3 * 5^3
%e 18 1296 = 6^4 = 2^4 * 3^4
%e 44 27000 = 30^3 = 2^3 * 3^3 * 5^3
%p N:= 30000: # for terms <= N
%p SF:= select(numtheory:-issqrfree, [seq(i,i=2..floor(N^(1/3)))]):
%p sort(map(proc(t) local i; seq(t^i,i=3..ilog[t](N)) end proc, SF)); # _Robert Israel_, Nov 11 2025
%t nn = 2^15; mm = Sqrt[nn]; i = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range[2, Sqrt[nn]], SquareFreeQ] ]; Union@ Reap[ While[j = 3; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 3, i++] ][[-1, 1]]
%o (Python)
%o from sympy import integer_nthroot
%o from oeis_sequences.OEISsequences import bisection, squarefreepi
%o def A390540(n):
%o def f(x): return n+x-sum(squarefreepi(integer_nthroot(x,k)[0])-1 for k in range(3,x.bit_length()))
%o return bisection(f,n,n) # _Chai Wah Wu_, Nov 11 2025
%o (PARI) isok(k) = ispower(k, , &m) && valuation(k, m) > 2 && issquarefree(m); \\ _Amiram Eldar_, Nov 11 2025
%Y Cf. A001597, A001694, A013929, A036966, A062503, A072777, A246547, A246549, A303606, A388304.
%Y Cf. A082020, A368250.
%K nonn,easy
%O 1,1
%A _Michael De Vlieger_, Nov 10 2025