A300858 a(n) = A243823(n) - A243822(n).

0, 0, 0, 0, 0, -1, 0, 1, 1, -1, 0, -1, 0, 1, 2, 4, 0, -1, 0, 3, 4, 3, 0, 3, 3, 5, 6, 7, 0, -5, 0, 11, 6, 7, 6, 6, 0, 9, 8, 11, 0, 1, 0, 13, 12, 13, 0, 13, 5, 13, 12, 17, 0, 13, 10, 19, 14, 19, 0, 5, 0, 21, 18, 26, 12, 11, 0, 23, 18, 15, 0, 25, 0, 25, 24, 27

Offset: 1

Author

Michael De Vlieger, Mar 14 2018

Keywords

• sign

Comments

Consider numbers in the cototient of n, listed in row n of A121998. For composite n > 4, there are nondivisors m in the cototient, listed in row n of A133995. Of these m, there are two species. The first are m that divide n^e with integer e > 1, while the last do not divide n^e. These are listed in row n of A272618 and A272619, and counted by A243822(n) and A243823(n), respectively. This sequence is the difference between the latter and the former species of nondivisors in the cototient of n.

Since A045763(n) = A243822(n) + A243823(n), this sequence examines the balance of the two components among nondivisors in the cototient of n.

For positive n < 6 and for p prime, a(n) = a(p) = 0, thus a(A046022(n)) = 0.

For prime powers p^e, a(p^e) = A243823(p^e), since A243822(p^e) = 0, thus a(n) = A243823(n) for n in A000961.

Value of a(n) is generally nonnegative; a(n) is negative for n = {6, 10, 12, 18, 30}; a(30) = -5, but a(n) = -1 for the rest of the aforementioned numbers. These five numbers are a subset of A295523.

Examples

			a(6) = -1 since the only nondivisor in the cototient of 6 is 4, which divides 6^e with e > 1; therefore 0 - 1 = -1.
a(8) = 1 since the only nondivisor in the cototient of 8 is 6, and 6 does not divide 8^e with e > 1, therefore 1 - 0 = 1.
Some values of a(n) and related sequences:
   n  a(n) A243823(n) A243822(n)    A272619(n)       A272618(n)
  -------------------------------------------------------------
   1   0          0          0      -                -
   2   0          0          0      -                -
   3   0          0          0      -                -
   4   0          0          0      -                -
   5   0          0          0      -                -
   6  -1          0          1      -                {4}
   7   0          0          0      -                -
   8   1          1          0      {6}              -
   9   1          1          0      {6}              -
  10  -1          1          2      {6}              {4,8}
  11   0          0          0      -                -
  12  -1          1          2      {10}             {8,9}
  13   0          0          0      -                -
  14   1          3          2      {6,10,12}        {4,8}
  15   2          3          1      {6,10,12}        {9}
  16   4          4          0      {6,10,12,14}     -
  17   0          0          0      -                -
  18  -1          3          4      {10,14,15}       {4,8,12,16}
  19   0          0          0      -                -
  20   3          5          2      {6,12,14,15,18}  {8,16}
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000010, A000961, A010846, A046022, A121998, A133995, A173540, A243822, A243823, A272618, A272619, A295523, A300859, A300861.

Programs

Mathematica

f[n_] := Count[Range@ n, _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)]; Array[#1 - #3 + 1 - 2 #2 + #4 & @@ {#, f@ #, EulerPhi@ #, DivisorSigma[0, #]} &, 76]

PARI

a(n) = 1 + n + numdiv(n) - eulerphi(n) - 2*sum(k=1, n, if(gcd(n,k)-1, 0, moebius(k)*(n\k))); \\ Michel Marcus, Mar 17 2018

Formula

a(n) = 1 + n - A000010(n) - 2*A010846(n) + A000005(n).