A376505 - OEIS

%I #10 Oct 02 2024 09:32:46

%S 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 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 0,5,0,7,0,0,2,1,0,8,0,4,0,0,0,11,0,0,0

%N Number of m <= n such that rad(m) | n that are neither squarefree nor prime powers, where rad = A007947.

%H Michael De Vlieger, <a href="/A376505/b376505.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 a(n) = A010846(n) - (Sum_{p|n} floor(log n / log p)) - 2^omega(n) + omega(n), where omega = A001221.

%F a(n) = A010846(n) - A361373(n) - A376504(n) + 1.

%F a(n) = 0 for n = p^k, where p is prime and k >= 0, i.e., n in A000961.

%F Intersection of A126706 and row n of A162306.

%e 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 a(k) = 0 for k < 12 since 12 is the smallest number that is neither squarefree nor prime powers.

%e a(12) = 1 since m = 12 is such that 12 <= 12 and rad(12) | 12.

%e a(18) = 2 since both k = 12 and k = 18 are such that rad(k) | 18.

%e 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 (* Load "theta" program from this <a href="https://oeis.org/A369609/a369609.txt">link</a> in A369609 *)

%t {0}~Join~Table[theta[n] - Total@ Map[Floor@ Log[#, n] &, #1] - 2^#2 + #2 & @@ {#, Length[#]} &@ FactorInteger[n][[All, 1]], {n, 2, 120}]

%o (PARI) rad(n) = factorback(factorint(n)[, 1]);

%o a(n) = sum(m=1, n, if (!issquarefree(m) && !isprimepower(m), ((n % rad(m))==0))); \\ _Michel Marcus_, Sep 29 2024

%Y 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 nonn,easy

%O 1,18

%A _Michael De Vlieger_, Sep 28 2024