cp's OEIS Frontend

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

%I A284258 #20 Mar 23 2021 05:38:41
%S A284258 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,
%T A284258 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,
%U A284258 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
%N A284258 a(n) = number of distinct prime factors of n that are > the square of smallest prime factor of n, a(1) = 0.
%H A284258 Antti Karttunen, <a href="/A284258/b284258.txt">Table of n, a(n) for n = 1..10001</a>
%F A284258 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.
%F A284258 a(n) = A001221(A284254(n)).
%F A284258 a(n) = A001221(n) - A284259(n).
%F A284258 a(n) <= A284256(n).
%e A284258 For n = 50, 2*5*5, the prime factor > 2^2 is 5, which is counted only once, thus a(50) = 1.
%e A284258 For n = 70, 2*5*7, the prime factors > 2^2 are 5 and 7, thus a(70) = 2.
%t A284258 Table[If[n == 1, 0, Count[#, d_ /; d > First[#]^2] &@ FactorInteger[n][[All, 1]]], {n, 120}] (* _Michael De Vlieger_, Mar 24 2017 *)
%o A284258 (Scheme) (define (A284258 n) (A001221 (A284254 n)))
%o A284258 (PARI)
%o A284258 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));
%o A284258 a(n) = if(A(n)==1, 1, A(n)*a(n/A(n)));
%o A284258 for(n=1, 150, print1(omega(a(n)),", ")) \\ _Indranil Ghosh_, after _David A. Corneth_, Mar 24 2017
%o A284258 (Python)
%o A284258 from sympy import primefactors
%o A284258 def omega(n): return len(primefactors(n))
%o A284258 def A(n):
%o A284258     for i in primefactors(n):
%o A284258         if i>min(primefactors(n))**2: return i
%o A284258     return 1
%o A284258 def a(n): return 1 if A(n)==1 else A(n)*a(n//A(n))
%o A284258 print([omega(a(n)) for n in range(1, 151)]) # _Indranil Ghosh_, Mar 24 2017
%Y A284258 Cf. A001221, A020639, A057427, A284252, A284253, A284254, A284255, A284257, A284259, A284261.
%Y A284258 Cf. A251726 (gives the positions of zeros after the initial a(1)=0).
%Y A284258 Differs from related A284256 for the first time at n=50, where a(50)=1, while A284256(50)=2.
%K A284258 nonn
%O A284258 1,70
%A A284258 _Antti Karttunen_, Mar 24 2017