cp's OEIS Frontend

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

%I A262202 #39 Oct 24 2015 11:50:24
%S A262202 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 A262202 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 A262202 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 A262202 Number of divisors d | n such that d^2 < n and d^2 does not divide n.
%C A262202 a(n) = 0 if n is a prime power.
%C A262202 From _Michael De Vlieger_, Sep 15 2015: (Start)
%C A262202 Let k be a divisor d | n such that d^2 < n and d^2 does not divide n.
%C A262202 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 A262202 a(n) <= A045763(n), as any k neither divides nor is coprime to n.
%C A262202 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 A262202 (End)
%C A262202 a(n) = 0 if and only if n is a prime power (A000961). - _Robert Israel_, Sep 22 2015
%C A262202 From _Robert Israel_, Oct 22 2015: (Start)
%C A262202 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 A262202 This appears to be "if and only if". (End)
%H A262202 Charles R Greathouse IV, <a href="/A262202/b262202.txt">Table of n, a(n) for n = 1..10000</a>
%e A262202 a(6) = 1 because (1, 4, 9, 36) are squares of divisors of 6 and only 4 is proper non-divisor of 6.
%p A262202 f:= n -> nops(select(t -> (t^2 < n) and (n mod t^2 <> 0), numtheory:-divisors(n))):
%p A262202 map(f, [$1..100]); # _Robert Israel_, Sep 22 2015
%t A262202 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 A262202 (PARI) a(n) = sumdiv(n, d, (d^2 < n) && (n % d^2)); \\ _Michel Marcus_, Sep 15 2015
%Y A262202 Cf. A000961, A006881, A010846, A045763, A054753, A085986, A243822, A143731.
%K A262202 nonn
%O A262202 1,24
%A A262202 _Juri-Stepan Gerasimov_, Sep 15 2015
%E A262202 Definition and a(80) corrected by _Charles R Greathouse IV_, Sep 15 2015