A394767 - OEIS

%I #15 Apr 11 2026 20:07:59

%S 0,0,0,0,1,5,15,33,74,161,346,724,1490,3045,6183,12504,25196,50675,

%T 101769,204111,409043,819280,1640260,3282967,6569393,13143658,

%U 26294336,52598680,105211612,210443477,420915409,841871271,1683800194,3367682134,6735480118,13471124232

%N Number of numbers k <= 2^n such that k is neither squareful nor squarefree.

%H Michael De Vlieger, <a href="/A394767/b394767.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = A000079(n) - A062762(n) - A143658(n) + 1.

%F a(n) = A372403(n) - A380431(n).

%F a(n) = 1 + 2^n - (Sum_{j=1..2^(n/3)} |mu(j)| * floor(sqrt(2^n/j^3))) - Sum_{k=1..2^(n/2)} mu(k) * floor((2^n)/k^2), where mu = A008683.

%e a(0)..a(3) = 0 since 12 is the smallest number in A332785.

%e a(4) = 1 since only 12 is no larger than 2^4 = 16.

%e a(5) = 5 since a(4) = 1, and additionally, 4 more numbers in A332785 are no larger than 2^5 = 32, these are {18, 20, 24, 28}.

%e a(6) = 15 since a(5) = 5, and additionally, 10 more are no larger than 2^6 = 64, i.e., {40, 44, 45, 48, 50, 52, 54, 56, 60, 63}, etc.

%t Table[1 + 2^n - Sum[Abs[MoebiusMu[j]]*Floor[Sqrt[(2^n)/j^3]], {j, 2^(n/3)}] - Sum[MoebiusMu[k]*Floor[(2^n)/k^2], {k, 2^(n/2)}], {n, 0, 30}]

%o (Python)

%o from math import isqrt

%o from sympy import integer_nthroot

%o from oeis_sequences.OEISsequences import squarefreepi

%o def A394767(n):

%o     m = 1<<n

%o     c, l, j = m+1-squarefreepi(integer_nthroot(m,3)[0])-squarefreepi(m), 0, isqrt(m)

%o     while j>1:

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

%o         w = squarefreepi(k2-1)

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

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

%o     return c+l # _Chai Wah Wu_, Apr 11 2026

%Y Cf. A000079, A001694, A005117, A062762, A143658, A332785, A372403, A380431.

%K nonn

%O 0,6

%A _Michael De Vlieger_, Apr 03 2026