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A376505
Number of m <= n such that rad(m) | n that are neither squarefree nor prime powers, where rad = A007947.
2
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
1,18
LINKS
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.
FORMULA
a(n) = A010846(n) - (Sum_{p|n} floor(log n / log p)) - 2^omega(n) + omega(n), where omega = A001221.
a(n) = A010846(n) - A361373(n) - A376504(n) + 1.
a(n) = 0 for n = p^k, where p is prime and k >= 0, i.e., n in A000961.
Intersection of A126706 and row n of A162306.
EXAMPLE
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.)
MATHEMATICA
(* Load "theta" program from this <a href="https://oeis.org/A369609/a369609.txt">link</a> in A369609 *)
{0}~Join~Table[theta[n] - Total@ Map[Floor@ Log[#, n] &, #1] - 2^#2 + #2 & @@ {#, Length[#]} &@ FactorInteger[n][[All, 1]], {n, 2, 120}]
PROG
(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
CROSSREFS
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).
Sequence in context: A187143 A187144 A123635 * A124304 A165408 A186733
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Sep 28 2024
STATUS
approved