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A284258
a(n) = number of distinct prime factors of n that are > the square of smallest prime factor of n, a(1) = 0.
11
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
1,70
LINKS
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).
EXAMPLE
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.
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 (A284258 n) (A001221 (A284254 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(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
CROSSREFS
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.
Sequence in context: A321936 A085491 A321013 * A322389 A336388 A277735
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 24 2017
STATUS
approved