A262202 Number of divisors d | n such that d^2 < n and d^2 does not divide n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 0, 3, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 1, 0, 1, 3, 0, 1, 1, 3, 0, 2, 0, 1, 1, 1, 1, 3, 0, 2, 0, 1, 0, 4, 1, 1, 1, 2, 0, 4, 1, 1, 1, 1, 1, 3, 0

Offset: 1

Author

Juri-Stepan Gerasimov, Sep 15 2015

Keywords

• nonn

Comments

a(n) = 0 if n is a prime power.

From Michael De Vlieger, Sep 15 2015: (Start)

Let k be a divisor d | n such that d^2 < n and d^2 does not divide n.

a(n) <= A010846(n), as any k is regular to n, i.e., k is a product less than n of the prime divisors of n.

a(n) <= A045763(n), as any k neither divides nor is coprime to n.

a(n) <= A243822(n), as any k is a "semidivisor" of n, i.e., k is a product less than n of the prime divisors of n that do not divide n.

(End)

a(n) = 0 if and only if n is a prime power (A000961). - Robert Israel, Sep 22 2015

From Robert Israel, Oct 22 2015: (Start)

a(n) = 1 if n = p^i * q^j where p and q are distinct primes and 1 <= i,j <= 2, i.e. n is in A006881, A054753 or A085986.

This appears to be "if and only if". (End)

Examples

			a(6) = 1 because (1, 4, 9, 36) are squares of divisors of 6 and only 4 is proper non-divisor of 6.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000961, A006881, A010846, A045763, A054753, A085986, A243822, A143731.

Programs

Maple

f:= n -> nops(select(t -> (t^2 < n) and (n mod t^2 <> 0), numtheory:-divisors(n))):
map(f, [$1..100]); # Robert Israel, Sep 22 2015

Mathematica

f[n_] := Block[{d = Divisors@ n}, Select[d^2, And[Mod[n, #] != 0, # < n] &]]; Length@ f@ # & /@ Range@ 120 (* Michael De Vlieger, Sep 15 2015 *)

PARI

a(n) = sumdiv(n, d, (d^2 < n) && (n % d^2)); \\ Michel Marcus, Sep 15 2015

Extensions

Definition and a(80) corrected by Charles R Greathouse IV, Sep 15 2015