A322818 - OEIS

%I #7 Dec 27 2018 17:36:37

%S 0,1,1,1,1,2,1,1,1,2,1,1,1,2,2,1,1,0,1,1,2,2,1,1,1,2,1,1,1,3,1,1,2,2,

%T 2,0,1,2,2,1,1,3,1,1,1,2,1,1,1,-1,2,1,1,-1,2,1,2,2,1,1,1,2,1,1,2,3,1,

%U 1,2,3,1,1,1,2,0,1,2,3,1,2,1,2,1,1,2,2,2,1,1,0,2,1,2,2,2,1,1,-1,1,-1,1,3,1,1,3

%N a(n) = A001222(n) - A001222(A285328(n)), where A285328(n) gives the next smaller m that has same prime factors as n (ignoring multiplicity), or 1 if n is squarefree, and A001222 gives the number of prime factors, when counted with multiplicity.

%H Antti Karttunen, <a href="/A322818/b322818.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = A001222(n) - A001222(A285328(n)).

%F a(A005117(n)) = A001222(A005117(n)).

%e For n = 6 = 2*3, there is no smaller number with only the prime factors 2 and 3 as 6 is squarefree, thus A285328(6) = 1, and a(6) = A001222(6) = 2.

%e For n = 40 = 2^3 * 5^1, the next smaller number with the same prime factors is 20 = 2^2 * 5^1. While 40 has 3+1 = 4 prime factors in total, 20 has 2+1 = 3, thus a(40) = 4-3 = 1.

%e For n = 50 = 2^1 * 5^2, the next smaller number with the same prime factors is 40 = 2^3 * 5^1, thus a(50) = (1+2)-(3+1) = -1.

%o (PARI)

%o A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947

%o A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); };

%o A322818(n) = (bigomega(n)-bigomega(A285328(n)));

%Y Cf. A001222, A005117, A285328, A322817.

%K sign

%O 1,6

%A _Antti Karttunen_, Dec 27 2018