Original entry on oeis.org
0, 1, 4, 18, 128, 1404, 18261, 310346, 5896727, 135624239, 3933101823, 121926157640, 4511267827531, 184961980943492, 7953365180610400, 373808163488684049, 19811832664899731265, 1168898127229083969892
Offset: 0
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Table[-1 + Sum[MoebiusMu[k]*Floor[#/(k^2)], {k, Floor[Sqrt[#]]}] &[Product[Prime[i], {i, n}]], {n, 0, 12}] (* Michael De Vlieger, Jan 24 2025 *)
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a(n) = my(t=vecprod(primes(n))-1); sum(i=1, sqrtint(t), t\i^2*moebius(i)); \\ Jinyuan Wang, Jan 24 2025
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from math import isqrt
from sympy import primorial, mobius
def A158341(n):
if n == 0: return 0
m = primorial(n)-1
return sum(mobius(k)*(m//k**2) for k in range(1,isqrt(m)+1)) # Chai Wah Wu, Jan 25 2025
A380430
Number of powerful numbers k that are not powers of primes (i.e., k is in A286708) that do not exceed the primorial number A002110(n).
Original entry on oeis.org
0, 0, 0, 0, 7, 50, 254, 1245, 5898, 29600, 163705, 925977, 5690175, 36681963, 241663896, 1662446097, 12134853382, 93406989325, 730785520398, 5990426525483, 50538885715526, 432266550168097, 3845700235189327, 35065304557027821, 334652745159828239, 3262707438761612651
Offset: 0
Let P = A002110 and let s = A286708 = A001694 \ A246547 \ {1}.
a(0..3) = 0 since the smallest number in s is 36.
a(4) = 7 since P(4) = 210 and numbers in s that are less than 210 include {36, 72, 100, 108, 144, 196, 200}, etc.
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Table[-1 + Sum[If[MoebiusMu[j] != 0, Floor[Sqrt[#/j^3]], 0], {j, #^(1/3)}] - Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor[Log2[#]]}] &[Product[Prime[i], {i, n}]], {n, 0, 12} ]
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from math import isqrt
from sympy import primorial, primepi, integer_nthroot, mobius
def A380430(n):
def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
if n == 0: return 0
m = primorial(n)
c, l, j = int(squarefreepi(integer_nthroot(m, 3)[0])-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))-1), 0, isqrt(m)
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-l # Chai Wah Wu, Feb 25 2025
A381391
Number of k <= 10^n that are neither squarefree nor prime powers (i.e., k is in A126706).
Original entry on oeis.org
0, 29, 367, 3866, 39098, 391838, 3920154, 39205902, 392069187, 3920718974, 39207261564, 392072817656, 3920728751139, 39207289143932, 392072896183208, 3920728975677128, 39207289797472001, 392072898095046811, 3920728981307675534, 39207289814141997459, 392072898144605471040
Offset: 1
Let S = A126706.
a(1) = 0 since the smallest term in S is 12.
a(2) = 29 since S(1..29) = {12, 18, 20, 24, ..., 99, 100}, etc.
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Table[10^n - Sum[PrimePi@ Floor[10^(n/k)], {k, 2, Floor[Log2[10^n]]}] - Sum[MoebiusMu[k]*Floor[10^n/(k^2)], {k, Floor[Sqrt[10^n]]}], {n, 10}]
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from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A381391(n):
m = 10**n
return int(-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))-sum(mobius(k)*(m//k**2) for k in range(2, isqrt(m)+1))) # Chai Wah Wu, Feb 23 2025
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