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Number of prime divisors p | n such that p^2 < n and p^2 does not divide n.
1

%I #18 Mar 20 2018 19:01:27

%S 0,0,0,0,0,1,0,0,0,1,0,1,0,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0,0,3,0,0,1,1,

%T 1,0,0,1,1,1,0,2,0,0,1,1,0,1,0,1,1,0,0,1,1,1,1,1,0,2,0,1,1,0,1,2,0,0,

%U 1,3,0,0,0,1,1,0,1,2,0,1,0,1,0,2,1,1,1,0,0,2,1,0,1,1,1,1,0

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

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

%C From _Michael De Vlieger_, Sep 17 2015: (Start)

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

%C a(n) <= A262202(n), as any k = p^2 is a special case of a (prime or nonprime) divisor d of n where d is prime.

%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)

%H Antti Karttunen, <a href="/A262262/b262262.txt">Table of n, a(n) for n = 1..65537</a>

%e a(6) = 1 because 4, 9 are squares of prime divisors of 6 and only 4 < 6 does not divide 6.

%t f[n_] := Block[{d = First /@ FactorInteger@ n}, Select[d^2, And[Mod[n, #] != 0, # < n] &]]; Length@ f@ # & /@ Range@ 120 (* _Michael De Vlieger_, Sep 17 2015 *)

%o (PARI) a(n) = sumdiv(n, d, isprime(d) && (d^2 < n) && (n % d^2)); \\ _Michel Marcus_, Sep 17 2015

%o (PARI) a(n)=my(f=factor(n)[,1]); sum(i=1,#f,f[i]^2<n && n%f[i]^2) \\ _Charles R Greathouse IV_, Sep 17 2015

%Y Cf. A010846, A045763, A243822, A262202.

%K nonn,easy

%O 1,30

%A _Juri-Stepan Gerasimov_, Sep 16 2015