This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380402 #15 Feb 25 2025 14:56:08
%S A380402 0,0,1,6,14,34,75,187,551,1954,8317,38582,200978,1125541,6562122,
%T A380402 40444003,266832233,1870169623,13424553758,101495825622,793832121165,
%U A380402 6325729776075,52616754936494,450157758564742,3999323787879764,37180986240914714,353667558431662474
%N A380402 Number of proper prime powers (in A246547) that do not exceed primorial A002110(n).
%F A380402 a(n) = Sum_{k=2..floor(log_2(P(n)))} pi(floor(P(n)^(1/k))), where P(n) = A002110(n).
%e A380402 Let s = A246547.
%e A380402 a(0) = a(1) = 0 since P(0) = 1 and P(1) = 2, and the smallest number in s is 4.
%e A380402 a(2) = 1 since P(2) = 6, and s(1) = 4 is the only term in s <= 6.
%e A380402 a(3) = 6 since P(3) = 30, and the set s(1..6) = {4, 8, 9, 16, 25, 27} contains k <= 30.
%e A380402 a(4) = 14 since P(4) = 210, and the set s(1..14) = {4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169} contains k <= 210, etc.
%t A380402 Table[Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor@ Log2[#]}] &[Product[Prime[i], {i, n}]], {n, 0, nn}]
%o A380402 (Python)
%o A380402 from sympy import primorial, primepi, integer_nthroot
%o A380402 def A380402(n):
%o A380402 if n == 0: return 0
%o A380402 m = primorial(n)
%o A380402 return int(sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))) # _Chai Wah Wu_, Jan 24 2025
%Y A380402 Cf. A036386, A246547.
%K A380402 nonn,hard
%O A380402 0,4
%A A380402 _Michael De Vlieger_, Jan 23 2025
%E A380402 a(24) corrected by _Chai Wah Wu_, Jan 25 2025
%E A380402 a(26) from _Jinyuan Wang_, Feb 25 2025