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The Dedekind psi function is defined by the formula
or equivalently
where as usual if and
Compare with Euler's totient function
Properties
The Dedekind psi function is a multiplicative arithmetic function, e.g.,
when
All values of the Dedekind psi function for are even.
Formulae
-
where is the sum of divisors of and is the radical (squarefree kernel) of .
Generating function
Dirichlet generating function
Related functions
Dedekind psi summatory function
The Dedekind psi summatory function (partial sums of the Dedekind psi function) is
where is Dedekind psi function.
All values of the Dedekind psi summatory function for are even.
Half difference of Dedekind psi function and Euler's totient function
n 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, ...
psi(n) 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, 72, ...
phi(n) 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, ...
psi(n) - phi(n) 0, 2, 2, 4, 2, 10, 2, 8, 6, 14, 2, 20, 2, 18, 16, 16, 2, 30, 2, 28, 20, 26, 2, 40, 10, 30, 18, 36, 2, 64, 2, 32, 28, 38, 24, 60, ...
(psi - phi)/2 0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20, 5, 15, 9, 18, 1, 32, 1, 16, 14, 19, 12, 30, ...
A069359 0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20, 5, 15, 9, 18, 1,*31,* 1, 16, 14, 19, 12, 30, ...
A003415 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, 15, 27, 32, 1, 31, 1, 80, 14, 19, 12, 60, ...
The sequence for is nearly identical, at least for small , but is NOT A069359!
The above data suggests that is 0 only for and 1 only when is prime. It also suggests that when is a prime power that we get in the sequence.
Product of the Dedekind psi function with Euler's totient function
The product of the Dedekind psi function with Euler's totient function gives
where is the squarefree kernel of .
Since is divisible by 24 (A024702) when is congruent to 1 or 5 modulo 6 and , we deduce that is divisible by if is divisible by neither 2 nor 3 or both 2 and 3, and is divisible by if is divisible by either (but not both) 2 or 3, being the number of distinct prime factors of .
Also the Jordan function J_2(n) (a generalization of phi(n))
- (A007434)
Also the Moebius transform of squares
Multiplicative with
- .
Quotient of the Dedekind psi function by Euler's totient function
The quotient of the Dedekind psi function by Euler's totient function gives
Sequences
A001615 Dedekind psi function
.
-
{1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, 72, 38, 60, 56, 72, 42, 96, 44, 72, 72, 72, 48, 96, 56, 90, 72, 84, 54, 108, 72, 96, 80, ...}
A173290 Dedekind psi summatory function.
-
{1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, 132, 156, 180, 204, 222, 258, 278, 314, 346, 382, 406, 454, 484, 526, 562, 610, 640, 712, 744, 792, 840, 894, 942, 1014, 1052, 1112, 1168, 1240, 1282, 1378, 1422, ...}
A??????
(Dedekind ψ (n) − Euler’s totient φ (n)) / 2 (is nearly identical, at least for small
, but is NOT
A069359!)
-
{0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20, 5, 15, 9, 18, 1, 32, 1, 16, 14, 19, 12, 30, ...}
A007434 Jordan function
.
-
{1, 3, 8, 12, 24, 24, 48, 48, 72, 72, 120, 96, 168, 144, 192, 192, 288, 216, 360, 288, 384, 360, 528, 384, 600, 504, 648, 576, 840, 576, 960, 768, 960, 864, 1152, 864, 1368, 1080, 1344, 1152, 1680, 1152, 1848, ...}
See also
External links