A262262 Number of prime divisors p | n such that p^2 < n and p^2 does not divide n.

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, 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, 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

Offset

1,30

Comments

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

From Michael De Vlieger, Sep 17 2015: (Start)

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

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.

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)

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Example

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A010846, A045763, A243822, A262202.

Sequence in context: A359270 A318508 A100655 * A079275 A236314 A236322

Adjacent sequences: A262259 A262260 A262261 * A262263 A262264 A262265

Keyword

nonn,easy

Author

Juri-Stepan Gerasimov, Sep 16 2015

Status

approved