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

1,2

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.

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.

Formula

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

Example

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}
  ...

Mathematica

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

Crossrefs

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

Sequence in context: A345874 A338720 A320013 * A175193 A353861 A073093

Adjacent sequences: A299987 A299988 A299989 * A299991 A299992 A299993

Keyword

sign

Author

Michael De Vlieger, Feb 25 2018

Status

approved