cp's OEIS Frontend

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).

%I A380430 #13 Feb 25 2025 13:13:43
%S A380430 0,0,0,0,7,50,254,1245,5898,29600,163705,925977,5690175,36681963,
%T A380430 241663896,1662446097,12134853382,93406989325,730785520398,
%U A380430 5990426525483,50538885715526,432266550168097,3845700235189327,35065304557027821,334652745159828239,3262707438761612651
%N 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).
%F A380430 a(n) = A380254(n) - A380402(n) - 1.
%F A380430 a(n) <= A380403(n) since A286708 is a proper subset of A126706.
%e A380430 Let P = A002110 and let s = A286708 = A001694 \ A246547 \ {1}.
%e A380430 a(0..3) = 0 since the smallest number in s is 36.
%e A380430 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.
%t A380430 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} ]
%o A380430 (Python)
%o A380430 from math import isqrt
%o A380430 from sympy import primorial, primepi, integer_nthroot, mobius
%o A380430 def A380430(n):
%o A380430     def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o A380430     if n == 0: return 0
%o A380430     m = primorial(n)
%o A380430     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)
%o A380430     while j>1:
%o A380430         k2 = integer_nthroot(m//j**2,3)[0]+1
%o A380430         w = squarefreepi(k2-1)
%o A380430         c += j*(w-l)
%o A380430         l, j = w, isqrt(m//k2**3)
%o A380430     return c-l # _Chai Wah Wu_, Feb 25 2025
%Y A380430 Cf. A001694, A126706, A246547, A286708, A380254, A380402.
%K A380430 nonn,hard
%O A380430 0,5
%A A380430 _Michael De Vlieger_, Jan 24 2025