A376505 Number of m <= n such that rad(m) | n that are neither squarefree nor prime powers, where rad = A007947.
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, 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, 0, 5, 0, 7, 0, 0, 2, 1, 0, 8, 0, 4, 0, 0, 0, 11, 0, 0, 0
Offset: 1
Examples
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.
a(k) = 0 for k < 12 since 12 is the smallest number that is neither squarefree nor prime powers.
a(12) = 1 since m = 12 is such that 12 <= 12 and rad(12) | 12.
a(18) = 2 since both k = 12 and k = 18 are such that rad(k) | 18.
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.)
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Hasse diagram of row 1440 of A162306 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.
Crossrefs
Programs
-
Mathematica
(* Load "theta" program from this A369609/a369609.txt">link in A369609 *) {0}~Join~Table[theta[n] - Total@ Map[Floor@ Log[#, n] &, #1] - 2^#2 + #2 & @@ {#, Length[#]} &@ FactorInteger[n][[All, 1]], {n, 2, 120}]
-
PARI
rad(n) = factorback(factorint(n)[, 1]); a(n) = sum(m=1, n, if (!issquarefree(m) && !isprimepower(m), ((n % rad(m))==0))); \\ Michel Marcus, Sep 29 2024