cp's OEIS Frontend

A380403 Number of integers k that are neither squarefree nor prime powers (in A126706) and that do not exceed primorial A002110(n).

%I A380403 #29 Feb 25 2025 14:54:50
%S A380403 0,0,0,5,67,871,11693,199976,3802411,87466676,2536583089,78634293907,
%T A380403 2909470106300,119288281458176,5129396144497507,241081619059363357,
%U A380403 12777325812023481231,753862222923258499554
%N A380403 Number of integers k that are neither squarefree nor prime powers (in A126706) and that do not exceed primorial A002110(n).
%F A380403 a(n) = P(n) - (Sum_{k=2..floor(log_2(P(n)))} pi(floor(P(n)^(1/k)))) - Sum_{k=1..floor(sqrt(P(n)))} mu(k)*floor(P(n)/(k^2)), where P(n) = A002110(n).
%F A380403 a(n) = A002110(n) - A380402(n) - A158341(n) - 1.
%e A380403 Let s = A126706 and let P(n) = A002110(n).
%e A380403 a(0..2) = 0 since P(0..2) = {1, 2, 6}, and the smallest number in s is 12.
%e A380403 a(3) = 5 since P(3) = 30, and the set s(1..6) = {12, 18, 20, 24, 28} contains k <= 30.
%e A380403 a(4) = 67 since P(4) = 210, and the set s(1..67) = {12, 18, 20, ..., 207, 208} contains k <= 210, etc.
%t A380403 Table[# - Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor[Log2[#]]}] - Sum[MoebiusMu[k]*Floor[#/(k^2)], {k, Floor[Sqrt[#]]}] &[Product[Prime[i], {i, n}]], {n, 0, 12}]
%o A380403 (Python)
%o A380403 from math import isqrt
%o A380403 from sympy import primorial, primepi, integer_nthroot, mobius
%o A380403 def A380403(n):
%o A380403     if n == 0: return 0
%o A380403     m = primorial(n)
%o A380403     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_, Jan 24 2025
%o A380403 (PARI) a(n) = my(q=vecprod(primes(n))); q - sum(k=2, logint(q, 2), primepi(sqrtnint(q, k))) - sum(k=1, sqrtint(q), q\k^2*moebius(k)); \\ _Jinyuan Wang_, Feb 25 2025
%Y A380403 Cf. A002110, A126706, A158341, A380402.
%K A380403 nonn,hard,more
%O A380403 0,4
%A A380403 _Michael De Vlieger_, Jan 23 2025
%E A380403 Offset changed to 0 by _Jinyuan Wang_, Jan 24 2025
%E A380403 a(16) from _Chai Wah Wu_, Jan 24 2025
%E A380403 a(17) from _Chai Wah Wu_, Jan 25 2025