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

1,6

Links

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

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.

Example

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.

Mathematica

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

Prog

(Scheme) (define (A284259 n) (A001221 (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(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

Crossrefs

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

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

Sequence in context: A331295 A325133 A236338 * A250068 A214566 A213982

Adjacent sequences: A284256 A284257 A284258 * A284260 A284261 A284262

Keyword

nonn

Author

Antti Karttunen, Mar 24 2017

Status

approved