A284258 a(n) = number of distinct prime factors of n that are > the square of smallest prime factor of n, a(1) = 0.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1

Offset: 1

Author

Antti Karttunen, Mar 24 2017

Keywords

• nonn

Examples

			For n = 50, 2*5*5, the prime factor > 2^2 is 5, which is counted only once, thus a(50) = 1.
For n = 70, 2*5*7, the prime factors > 2^2 are 5 and 7, thus a(70) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10001

Crossrefs

Cf. A001221, A020639, A057427, A284252, A284253, A284254, A284255, A284257, A284259, A284261.

Cf. A251726 (gives the positions of zeros after the initial a(1)=0).

Differs from related A284256 for the first time at n=50, where a(50)=1, while A284256(50)=2.

Programs

Mathematica

Table[If[n == 1, 0, Count[#, d_ /; d > First[#]^2] &@ FactorInteger[n][[All, 1]]], {n, 120}] (* Michael De Vlieger, Mar 24 2017 *)

PARI

A(n) = if(n<2, return(1), my(f=factor(n)[, 1]); for(i=2, #f, if(f[i]>f[1]^2, return(f[i]))); return(1));
a(n) = if(A(n)==1, 1, A(n)*a(n/A(n)));
for(n=1, 150, print1(omega(a(n)),", ")) \\ Indranil Ghosh, after David A. Corneth, Mar 24 2017

Python

from sympy import primefactors
def omega(n): return len(primefactors(n))
def A(n):
    for i in primefactors(n):
        if i>min(primefactors(n))**2: return i
    return 1
def a(n): return 1 if A(n)==1 else A(n)*a(n//A(n))
print([omega(a(n)) for n in range(1, 151)]) # Indranil Ghosh, Mar 24 2017

Scheme

(define (A284258 n) (A001221 (A284254 n)))

Formula

a(n) = Sum_{p|n, p prime and > spf(n)^2} sign(p), where spf(n) (A020639) gives the smallest prime factor of n, and sign(p) = A057427(p) = 1 for all p.

a(n) = A001221(A284254(n)).

a(n) = A001221(n) - A284259(n).

a(n) <= A284256(n).