A293460 a(n) = Sum_{k=1..n} sign(omega(n+1) - omega(n)) (where omega(m) = A001221(m), the number of distinct primes dividing m).

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 1, 2, 3, 2, 3, 3, 4, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 4, 4, 4

Offset: 0

Author

Rémy Sigrist, Oct 26 2017

Keywords

• sign

Comments

The sign function is defined by:

- sign(0) = 0,

- sign(n) = +1 for any n > 0,

- sign(n) = -1 for any n < 0.

a(n) corresponds to the number of integers up to n in A294277 minus the number of integers up to n in A294278.

The first negative value occurs at a(178) = -1.

Will this sequence change sign indefinitely?

Examples

			The following table shows the first terms of the sequence, alongside sign(omega(n+1)-omega(n)), omega(n+1) and omega(n):
n       a(n)    sign    w(n+1)  w(n)
-       ----    ----    ------  ----
0       0
1       1       1       1       0
2       1       0       1       1
3       1       0       1       1
4       1       0       1       1
5       2       1       2       1
6       1       -1      1       2
7       1       0       1       1
8       1       0       1       1
9       2       1       2       1
10      1       -1      1       2
11      2       1       2       1
12      1       -1      1       2
13      2       1       2       1
14      2       0       2       2
15      1       -1      1       2
16      1       0       1       1
17      2       1       2       1
18      1       -1      1       2
19      2       1       2       1
20      2       0       2       2
		

Links

• Georg Fischer, Table of n, a(n) for n = 0..1000
• Rémy Sigrist, Line graph of the first 10000 terms
• Rémy Sigrist, Line graph of the first 100000000 terms
• Rémy Sigrist, Line graph of the first 1000000000 terms
• Rémy Sigrist, Line graph of the first 10000000000 terms

Crossrefs

Cf. A001221, A006049, A294277, A294278.

Programs

Mathematica

Accumulate[Join[{0},Sign[Differences[PrimeNu[Range[90]]]]]] (* Harvey P. Dale, Dec 25 2024 *)

PARI

s = 0; for (n=1, 87, print1 (s ", "); s += sign(omega(n+1)-omega(n)))

Formula

a(0) = 0, and for any n > 0:

- a(A294277(n)) = a(A294277(n)-1) + 1,

- a(A006049(n)) = a(A006049(n)-1),

- a(A294278(n)) = a(A294278(n)-1) - 1.

Also: a(n) = #{ k / A294277(k) <= n } - #{ k / A294278(k) <= n }.