A322818 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.
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, 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, 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
Offset: 1
Keywords
Examples
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. 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. 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.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..20000