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%I #12 Mar 03 2018 14:58:18
%S -1,-2,-2,-3,-2,-3,-2,-4,-3,-2,-2,-4,-2,-2,-3,-5,-2,-2,-2,-4,-3,-1,-2,
%T -5,-3,-1,-4,-4,-2,2,-2,-6,-2,0,-3,-4,-2,0,-2,-5,-2,3,-2,-3,-4,0,-2,
%U -5,-3,0,-2,-3,-2,0,-3,-5,-2,0,-2,2,-2,0,-4,-7,-3,6,-2,-2
%N a(n) = A243822(n) - A000005(n).
%C Since A010846(n) = A000005(n) + A243822(n), this sequence examines the balance of the two components among "regular" numbers.
%C Value of a(n) is generally less frequently negative as n increases.
%C a(1) = -1.
%C For primes p, a(p) = -2 since 1 | p and the cototient is restricted to the divisor p.
%C 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.
%C 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).
%C For m >= 3, a(A002110(m)) is positive.
%C For m >= 9, a(A244052(m)) is positive.
%H Michael De Vlieger, <a href="/A299990/b299990.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="/A299990/a299990_1.txt">Examination of the relationships of the species of numbers enumerated in A010846</a>.
%F a(n) = A010846(n) - 2*A000005(n).
%e 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.
%e 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.
%e Some values of a(n) and related sequences:
%e n a(n) A010846(n) A243822(n) A000005(n) A272618(n)
%e ----------------------------------------------------
%e 1 -1 1 0 1 0
%e 2 -2 2 0 2 0
%e 3 -2 2 0 2 0
%e 4 -3 3 0 3 0
%e 5 -2 2 0 2 0
%e 6 -3 5 1 4 {4}
%e 7 -2 2 0 2 0
%e 8 -4 4 0 4 0
%e 9 -3 3 0 3 0
%e 10 -2 6 2 4 {4,8}
%e 11 -2 2 0 2 0
%e 12 -4 8 2 6 {8,9}
%e ...
%e 30 2 18 10 8 {4,8,9,12,16,18,20,24,25,27}
%e ...
%e 34 0 8 4 4 {4,8,16,32}
%e ...
%t Table[Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] - 2 DivisorSigma[0, n], {n, 68}]
%Y Cf. A000005, A002110, A010846, A243822, A272618, A272619, A299991, A299992, A300155, A300156, A300157.
%K sign
%O 1,2
%A _Michael De Vlieger_, Feb 25 2018
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