

A243822


Number of "semidivisors" of n, numbers m < n that do not divide n but divide n^e for some integer e > 1.


16



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
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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

Michael De Vlieger, Table of n, a(n) for n = 1..10000
M. De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 412.
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

Cf. A045763, A010846, A000005, A243823.
Sequence in context: A137899 A293516 A293898 * A277150 A156596 A282570
Adjacent sequences: A243819 A243820 A243821 * A243823 A243824 A243825


KEYWORD

nonn


AUTHOR

Michael De Vlieger, Jun 11 2014


STATUS

approved



