A284257 a(n) = number of prime factors of n that are < the square of smallest prime factor of n (counted with multiplicity), a(1) = 0.

0, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 1, 2, 4, 1, 3, 1, 2, 2, 1, 1, 4, 2, 1, 3, 2, 1, 2, 1, 5, 1, 1, 2, 4, 1, 1, 1, 3, 1, 2, 1, 2, 3, 1, 1, 5, 2, 1, 1, 2, 1, 4, 2, 3, 1, 1, 1, 3, 1, 1, 3, 6, 2, 2, 1, 2, 1, 1, 1, 5, 1, 1, 3, 2, 2, 2, 1, 4, 4, 1, 1, 3, 2, 1, 1, 3, 1, 3, 2, 2, 1, 1, 2, 6, 1, 1, 2, 2, 1, 2, 1, 3, 3, 1, 1, 5, 1, 1, 1, 4, 1, 2, 2, 2, 2, 1, 2, 4

Offset

1,4

Links

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

Formula

a(n) = A001222(A284255(n)).

a(n) = A001222(n) - A284256(n).

Example

For n = 45 = 3*3*5, all prime factors 3, 3 and 5 are less than 3^2, thus a(45) = 3.
For n = 120 = 2*2*2*3*5, the prime factors less than 2^2 are 2*2*2*3, thus a(120) = 4.

Mathematica

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

Prog

(Scheme) (define (A284257 n) (A001222 (A284255 n)))
(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(bigomega(n/a(n)), ", ")) \\ Indranil Ghosh, Mar 24 2017, after David A. Corneth
(Python)
from sympy import primefactors
def Omega(n): return 0 if n==1 else Omega(n//min(primefactors(n))) + 1
def A(n):
    pf = primefactors(n)
    if pf: min_pf2 = min(pf)**2
    for i in pf:
        if i > min_pf2: 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

Crossrefs

Cf. A001222, A020639, A284252, A284253, A284254, A284256, A284258, A284259.

Sequence in context: A322872 A356227 A356228 * A318883 A356233 A353382

Adjacent sequences: A284254 A284255 A284256 * A284258 A284259 A284260

Keyword

nonn

Author

Antti Karttunen, Mar 24 2017

Status

approved