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A262202 Number of divisors d | n such that d^2 < n and d^2 does not divide n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,24

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)

LINKS

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

EXAMPLE

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

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A275851 A067432 A192174 * A284413 A323879 A129308

Adjacent sequences:  A262199 A262200 A262201 * A262203 A262204 A262205

KEYWORD

nonn

AUTHOR

Juri-Stepan Gerasimov, Sep 15 2015

EXTENSIONS

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

STATUS

approved

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Last modified July 15 16:29 EDT 2019. Contains 325049 sequences. (Running on oeis4.)