A381496 Number of powerful numbers that are not prime powers that do not exceed 10^n.
0, 0, 3, 28, 133, 510, 1790, 5997, 19639, 63541, 204037, 652173, 2078320, 6609816, 20993381, 66612867, 211222374, 669428537, 2120835892, 6717184256, 21270247404, 67341572823, 213173925948, 674739560651, 2135491756895, 6758117426102, 21385762133815, 67670426242420
Offset: 0
Keywords
Examples
Let S = A286708 = A001694 \ A246547 = A126706 \ A001694.
a(0) = a(1) = 0 since 36 is the smallest term in S.
a(2) = 3 since S(1..3) = {36, 72, 100}.
a(3) = 28 since S(4..28) = {108, 144, ..., 972, 1000}.
a(4) = 133 since S(29..133) = {1089, 1125, ..., 9801, 10000}, etc.
Programs
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Mathematica
Table[Sum[Boole[SquareFreeQ[k]]*Floor[Sqrt[10^n/k^3]], {k, 10^(n/3)}] - Sum[PrimePi[10^(n/k)], {k, 2, n*Log2[10]}] - 1, {n, 0, 12}] -
Python
from math import isqrt from sympy import primepi, integer_nthroot, mobius def A381496(n): def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1))) m, l = 10**n, 0 j, c = isqrt(m), -1-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length())), while j>1: k2 = integer_nthroot(m//j**2,3)[0]+1 w = squarefreepi(k2-1) c += j*(w-l) l, j = w, isqrt(m//k2**3) return c+squarefreepi(integer_nthroot(m,3)[0])-l # Chai Wah Wu, Feb 25 2025
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