A389947 - OEIS

%I #33 Jan 12 2026 04:21:06

%S 144,324,400,576,784,1600,1936,2025,2304,2500,2704,2916,3136,3600,

%T 3969,4624,5625,5776,6400,7056,7744,8100,8464,9216,9604,9801,10816,

%U 12544,13456,13689,14400,15376,15876,17424,18225,18496,19600,21609,21904,22500,23104,23409,24336,25600,26244,26896,28224,29241,29584,30625,30976,32400

%N Squares of nonsquarefree numbers k that are not squareful.

%C Numbers in A386762 that are not cubefull.

%C Numbers in A389864 that are not cubefull, where A389864 is the union of A386762 and A383394, since the latter is a proper subset of A036966.

%C Intersection of A386762 and A362147 = intersection of A389864 and A362147.

%C Squares of terms in A332785. - _Chai Wah Wu_, Dec 01 2025

%C Superset of A392069; a(52) = 32400 is not in A392069.

%H Michael De Vlieger, <a href="/A389947/b389947.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%F From _Amiram Eldar_, Nov 23 2025: (Start)

%F Equals (A000290 \ A374291) \ A062503.

%F Sum_{n>=1} 1/a(n) = 1 + Pi^2/6 - 35745/(1382*Pi^2) = 0.02429317734362177637... . (End)

%e Table of n, a(n) for select n:

%e  n    a(n)

%e -----------------------------------

%e  1    144 = 12^2 = 2^4 *  3^2

%e  2    324 = 18^2 = 2^2 *  3^4

%e  3    400 = 20^2 = 2^4 *  5^2

%e  4    576 = 24^2 = 2^6 *  3^2

%e  5    784 = 28^2 = 2^4 *  7^2

%e  6   1600 = 40^2 = 2^6 *  5^2

%e  7   1936 = 44^2 = 2^4 * 11^2

%e  8   2025 = 45^2 = 3^4 *  5^2

%e  9   2304 = 48^2 = 2^8 *  3^2

%e 10   2500 = 50^2 = 2^2 *  5^4

%e 11   2704 = 52^2 = 2^4 * 13^2

%e 14   3600 = 60^2 = 2^4 *  3^2 * 5^2

%t Select[Range[155], And[#1 > 1, 0 < #2 < #1] & @@ {Length[#], Count[#[[;; , -1]], 1]} &[FactorInteger[#]] &]^2

%o (Python)

%o from math import isqrt

%o from sympy import integer_nthroot

%o from oeis_sequences.OEISsequences import bisection, squarefreepi

%o def A389947(n):

%o     def f(x):

%o         c, l, y = n+x-1, 0, isqrt(x)

%o         j = isqrt(y)

%o         while j>1:

%o             k2 = integer_nthroot(y//j**2,3)[0]+1

%o             w = squarefreepi(k2-1)

%o             c += j*(w-l)

%o             l, j = w, isqrt(y//k2**3)

%o         return c-y-l+squarefreepi(integer_nthroot(y,3)[0])+squarefreepi(y)

%o     return bisection(f,n,n) # _Chai Wah Wu_, Dec 01 2025

%Y Cf. A000290, A001597, A001694, A013929, A024619, A036966, A062503, A126706, A131605, A286708, A332785, A362147, A374291, A383394, A386762, A389864, A392069.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Nov 17 2025