A284259 - OEIS

%I #24 Mar 27 2021 06:17:05

%S 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,

%T 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,

%U 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

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

%H Antti Karttunen, <a href="/A284259/b284259.txt">Table of n, a(n) for n = 1..10001</a>

%F 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 a(n) = A001221(A284255(n)).

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

%F a(n) <= A284257(n).

%F a(A284262(n)) = n.

%e For n = 4 = 2*2, the prime factor 2 is less than 2^2, and it is counted only once, thus a(4) = 1.

%e For n = 45 = 3*3*5, both prime factors 3 and 5 are less than 3^2, thus a(45) = 2.

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

%o (Scheme) (define (A284259 n) (A001221 (A284255 n)))

%o (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));

%o a(n) = if(A(n)==1, 1, A(n)*a(n/A(n)));

%o for(n=1, 150, print1(omega(n/a(n)),", ")) \\ _Indranil Ghosh_, after _David A. Corneth_, Mar 24 2017

%o (Python)

%o from sympy import primefactors

%o def omega(n): return len(primefactors(n))

%o def A(n):

%o     for i in primefactors(n):

%o         if i>min(primefactors(n))**2: return i

%o     return 1

%o def a(n): return 1 if A(n)==1 else A(n)*a(n//A(n))

%o print([omega(n//a(n)) for n in range(1, 151)]) # _Indranil Ghosh_, Mar 24 2017

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

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

%K nonn

%O 1,6

%A _Antti Karttunen_, Mar 24 2017