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A243822 Number of "semidivisors" of n, numbers m < n that do not divide n but divide n^e for some integer e > 1. 33
0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 4, 0, 2, 1, 3, 0, 3, 0, 3, 0, 2, 0, 10, 0, 0, 2, 4, 1, 5, 0, 4, 2, 3, 0, 11, 0, 3, 2, 4, 0, 5, 0, 6, 2, 3, 0, 8, 1, 3, 2, 4, 0, 14, 0, 4, 2, 0, 1, 14, 0, 4, 2, 12, 0, 6, 0, 5, 3, 4, 1, 15, 0, 4, 0, 5, 0, 16, 1, 5, 3, 3, 0, 20, 1, 4, 3, 5, 1, 8, 0, 7, 2, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,10
COMMENTS
Semidivisors m < n are products of primes restricted to the prime divisors of n, however they have multiplicities of at least one prime divisor that exceeds the multiplicity of the corresponding prime divisor in n. As regular numbers in base n, the unit fractions of semidivisors have terminating expansions in base n (see Hardy & Wright). Semidivisors must be m < n, while the set of regular numbers in base n include the sets of semidivisors and divisors and can be larger than n.
a(n) = 0 for each composite n that are perfect prime powers p^e, since any semidivisor m must be p^a, with a < e, and all such possible p^a divide p^e.
a(n) > 0 for all composites n that are not perfect prime powers, since all squarefree semiprimes n = p * (p + 2) must have at least p^2 as an m that does not divide n. This is because p < sqrt(n), regardless of the magnitude of n. Further prime factors of n only reduce the relative size of the minimum p, ensuring that there will be a p_min^2 that does not divide n.
LINKS
M. De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
M. De Vlieger, Neutral Numbers
FORMULA
a(n) = A010846(n) - A000005(n).
a(n) = A045763(n) - A243823(n).
a(n) = (Sum_{1<=k<=n, gcd(n,k)=1} mu(k)*floor(n/k)) - tau(n). - Michael De Vlieger, May 10 2016, after Benoit Cloitre at A010846.
EXAMPLE
For n = 10 with prime divisors {2, 5}, the numbers {1, 2, 4, 5, 8, 10} are regular (cf. A010846). Subtract the set of divisors {1, 2, 5, 10} to obtain the set of semidivisors of 10, {4, 8}. So a(10) = 2.
Note that 4 divides 10^2 and 8 divides 10^3. The set of numbers less than 10 that are neither divisors nor totatives is {4, 6, 8}; 6 is a semitotative of 10, while {4, 8} are semidivisors.
MATHEMATICA
f[n_] := Block[{g, a}, g[x_] := First /@ FactorInteger@ x; a = g@ n; Length@ Select[Select[Range@ n, Complement[g@ #, a] == {} &], LCM[#, n] != n &]]; f /@ Range@ 120 (* Michael De Vlieger, Sep 15 2015 *)
Table[Total[MoebiusMu[#] Floor[n/#] &@ Select[Range@ n, CoprimeQ[#, n] &]] - DivisorSigma[0, n], {n, 120}] (* Michael De Vlieger, May 10 2016, faster *)
CROSSREFS
Sequence in context: A368817 A324301 A361235 * A277150 A156596 A282570
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Jun 11 2014
STATUS
approved

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Last modified March 18 22:34 EDT 2024. Contains 370951 sequences. (Running on oeis4.)