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

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

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

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

Example

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.

Prog

(PARI)
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
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)); };
A322818(n) = (bigomega(n)-bigomega(A285328(n)));

Crossrefs

Cf. A001222, A005117, A285328, A322817.

Sequence in context: A354999 A136177 A212178 * A362611 A353507 A362613

Adjacent sequences: A322815 A322816 A322817 * A322819 A322820 A322821

Keyword

sign

Author

Antti Karttunen, Dec 27 2018

Status

approved