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

0,6

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?

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

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 }.

Example

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

Prog

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

Crossrefs

Cf. A001221, A006049, A294277, A294278.

Sequence in context: A302031 A371883 A237353 * A231813 A158210 A322307

Adjacent sequences: A293457 A293458 A293459 * A293461 A293462 A293463

Keyword

sign

Author

Rémy Sigrist, Oct 26 2017

Status

approved