A262202 - OEIS

%I #39 Oct 24 2015 11:50:24

%S 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,

%T 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,

%U 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

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

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

%C From _Michael De Vlieger_, Sep 15 2015: (Start)

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

%C 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.

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

%C 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.

%C (End)

%C a(n) = 0 if and only if n is a prime power (A000961). - _Robert Israel_, Sep 22 2015

%C From _Robert Israel_, Oct 22 2015: (Start)

%C 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.

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

%H Charles R Greathouse IV, <a href="/A262202/b262202.txt">Table of n, a(n) for n = 1..10000</a>

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

%p f:= n -> nops(select(t -> (t^2 < n) and (n mod t^2 <> 0), numtheory:-divisors(n))):

%p map(f, [$1..100]); # _Robert Israel_, Sep 22 2015

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

%o (PARI) a(n) = sumdiv(n, d, (d^2 < n) && (n % d^2)); \\ _Michel Marcus_, Sep 15 2015

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

%K nonn

%O 1,24

%A _Juri-Stepan Gerasimov_, Sep 15 2015

%E Definition and a(80) corrected by _Charles R Greathouse IV_, Sep 15 2015