A299990 a(n) = A243822(n) - A000005(n).

-1, -2, -2, -3, -2, -3, -2, -4, -3, -2, -2, -4, -2, -2, -3, -5, -2, -2, -2, -4, -3, -1, -2, -5, -3, -1, -4, -4, -2, 2, -2, -6, -2, 0, -3, -4, -2, 0, -2, -5, -2, 3, -2, -3, -4, 0, -2, -5, -3, 0, -2, -3, -2, 0, -3, -5, -2, 0, -2, 2, -2, 0, -4, -7, -3, 6, -2, -2

Offset: 1

Author

Michael De Vlieger, Feb 25 2018

Keywords

• sign

Comments

Since A010846(n) = A000005(n) + A243822(n), this sequence examines the balance of the two components among "regular" numbers.

Value of a(n) is generally less frequently negative as n increases.

a(1) = -1.

For primes p, a(p) = -2 since 1 | p and the cototient is restricted to the divisor p.

For perfect prime powers p^e, a(p^e) = -(e + 1), since all m < p^e in the cototient of p^e that do not have a prime factor q coprime to p^e are powers p^k with 1 < p^k <= p^e; all such p^k divide p^e.

Generally for n with A001221(n) = 1, a(n) = -1 * A000005(n), since the cototient is restricted to divisors, and in the case of p^e > 4, divisors and numbers in A272619(p^e) not counted by A010846(p^e).

For m >= 3, a(A002110(m)) is positive.

For m >= 9, a(A244052(m)) is positive.

Examples

			a(6) = -3 since 6 has 4 divisors, and 4 | 6^2; A243822(6) = 1 and A000005(6) = 4; 1 - 4 = -3. Alternatively, A010846(6) = 5; 5 - 2*4 = -3.
a(30) = 2 since 30 has 8 divisors and the numbers {4, 8, 9, 12, 16, 18, 20, 24, 25, 27} divide 30^e with e > 1; A243822(30) = 10 and A000005(30) = 8; 10 - 8 = 2. Alternatively, A010846(30) = 18; 18 - 2*8 = 2.
Some values of a(n) and related sequences:
   n  a(n) A010846(n) A243822(n) A000005(n) A272618(n)
  ----------------------------------------------------
   1   -1          1          0          1  0
   2   -2          2          0          2  0
   3   -2          2          0          2  0
   4   -3          3          0          3  0
   5   -2          2          0          2  0
   6   -3          5          1          4  {4}
   7   -2          2          0          2  0
   8   -4          4          0          4  0
   9   -3          3          0          3  0
  10   -2          6          2          4  {4,8}
  11   -2          2          0          2  0
  12   -4          8          2          6  {8,9}
  ...
  30    2         18         10          8  {4,8,9,12,16,18,20,24,25,27}
  ...
  34    0          8          4          4  {4,8,16,32}
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Examination of the relationships of the species of numbers enumerated in A010846.

Crossrefs

Cf. A000005, A002110, A010846, A243822, A272618, A272619, A299991, A299992, A300155, A300156, A300157.

Programs

Mathematica

Table[Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] - 2 DivisorSigma[0, n], {n, 68}]

Formula

a(n) = A010846(n) - 2*A000005(n).