A284259 a(n) = number of distinct prime factors of n that are < the square of smallest prime factor of n, a(1) = 0.

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2

Offset: 1

Author

Antti Karttunen, Mar 24 2017

Keywords

• nonn

Examples

			For n = 4 = 2*2, the prime factor 2 is less than 2^2, and it is counted only once, thus a(4) = 1.
For n = 45 = 3*3*5, both prime factors 3 and 5 are less than 3^2, thus a(45) = 2.
		

Links

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

Crossrefs

Cf. A001221, A020639, A057427, A284252, A284253, A284254, A284255, A284257, A284258, A284260.

Cf. A284262 (where obtains first time value n, also positions of records).

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(n/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(n//a(n)) for n in range(1, 151)]) # Indranil Ghosh, Mar 24 2017

Scheme

(define (A284259 n) (A001221 (A284255 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(A284255(n)).

a(n) = A001221(n) - A284258(n).

a(n) <= A284257(n).

a(A284262(n)) = n.