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 A376505 #10 Oct 02 2024 09:32:46
%S A376505 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,1,0,0,0,3,0,0,0,1,0,4,0,0,0,0,
%T A376505 0,4,0,0,0,2,0,5,0,1,1,0,0,5,0,3,0,1,0,6,0,2,0,0,0,11,0,0,1,0,0,7,0,1,
%U A376505 0,5,0,7,0,0,2,1,0,8,0,4,0,0,0,11,0,0,0
%N A376505 Number of m <= n such that rad(m) | n that are neither squarefree nor prime powers, where rad = A007947.
%H A376505 Michael De Vlieger, <a href="/A376505/b376505.txt">Table of n, a(n) for n = 1..10000</a>
%H A376505 Michael De Vlieger, <a href="https://oeis.org/A376504/a376504.png">Hasse diagram of row 1440 of A162306</a> showing 4 squarefree composites in green, 3 primes in red, the empty product in gray, 17 perfect powers of primes in yellow, and 72 numbers that are neither squarefree nor prime powers in blue and purple, with purple additionally representing powerful numbers that are not prime powers.
%F A376505 a(n) = A010846(n) - (Sum_{p|n} floor(log n / log p)) - 2^omega(n) + omega(n), where omega = A001221.
%F A376505 a(n) = A010846(n) - A361373(n) - A376504(n) + 1.
%F A376505 a(n) = 0 for n = p^k, where p is prime and k >= 0, i.e., n in A000961.
%F A376505 Intersection of A126706 and row n of A162306.
%e A376505 a(2) = a(4) = a(p^k) = 0 since numbers m <= p^k such that rad(m) | p^k are all divisors that are prime powers p^j, j = 0..k.
%e A376505 a(k) = 0 for k < 12 since 12 is the smallest number that is neither squarefree nor prime powers.
%e A376505 a(12) = 1 since m = 12 is such that 12 <= 12 and rad(12) | 12.
%e A376505 a(18) = 2 since both k = 12 and k = 18 are such that rad(k) | 18.
%e A376505 a(30) = 4 since row 30 of A162306 has 4 numbers that are neither squarefree nor prime powers: {1, 2, 3, 4, 5, 6, 8, 9, 10, [12], 15, 16, [18], [20], [24], 25, 27, 30}, indicated by brackets. (The bracketed numbers happen to be the first 4 terms of A126706.)
%t A376505 (* Load "theta" program from this <a href="https://oeis.org/A369609/a369609.txt">link</a> in A369609 *)
%t A376505 {0}~Join~Table[theta[n] - Total@ Map[Floor@ Log[#, n] &, #1] - 2^#2 + #2 & @@ {#, Length[#]} &@ FactorInteger[n][[All, 1]], {n, 2, 120}]
%o A376505 (PARI) rad(n) = factorback(factorint(n)[, 1]);
%o A376505 a(n) = sum(m=1, n, if (!issquarefree(m) && !isprimepower(m), ((n % rad(m))==0))); \\ _Michel Marcus_, Sep 29 2024
%Y A376505 Cf. A000005, A000961, A001221, A010846, A126706, A162306, A376504, A361373 (intersection of A246655 and row n of A162306), A376504 (intersection of A120944 and row n of A162306).
%K A376505 nonn,easy
%O A376505 1,18
%A A376505 _Michael De Vlieger_, Sep 28 2024