A238230 Numbers m such that if x = sigma(m)-phi(m)-tau(m)-m then m = sigma(x)-phi(x)-tau(x)-x.

198, 294, 16008, 22232, 150030, 195320, 200274, 3038720, 12190720, 124904790, 167179722, 347943288, 426853240, 528995656, 646186568, 3588779502, 4798752860, 5376246738, 5898361924, 158380893880, 189740533470, 196271084296, 240458641570, 375653406648

Offset: 1

Author

Paolo P. Lava, Feb 20 2014

Keywords

• nonn
• more

Comments

All numbers of the form 2^k*5*p, where p = (7*2^k-2*k-5)/3 is prime, are fixed points and thus terms. This happens for k = 9, 10, 124, 352, 1468, 3339, 4365,... - Giovanni Resta, Mar 26 2014

The fixed points (terms with x = m) are 198, 294, 195320, 3038720, 12190720, ... - Amiram Eldar, Mar 31 2019

a(20) > 10^11. - Giovanni Resta, Apr 04 2019

Examples

			Fixed points: 198, 294, 195320,...
sigma(16008) = 43200, phi(16008) = 4928, tau(16008) = 32 and 43200 - 4928 - 32- 16008 = 22232.
sigma(22232) = 47760, phi(22232) = 9504, tau(22232) = 16 and 47760 - 9504 - 16 - 22232 = 16008.
		

Crossrefs

Cf. A000005, A000010, A000203, A055971, A238225-A238229.

Programs

Maple

with(numtheory); P:=proc(q)local a,n;
for n from 1 to q do a:=sigma(n)-phi(n)-tau(n)-n;
if a>0 and sigma(a)-phi(a)-tau(a)-a=n then print(n);
fi; od; end: P(10^6);

Mathematica

f[n_] := If[n > 0, DivisorSigma[1, n] - EulerPhi[n] - DivisorSigma[0, n] - n, 0]; s={}; Do[ If[f[f[n]] == n, AppendTo[s, n]], {n, 1, 200000}]; s (* Amiram Eldar, Mar 31 2019 *)

Extensions

a(8)-a(15) from Giovanni Resta, Mar 26 2014

a(16)-a(19) from Amiram Eldar, Mar 31 2019

a(19) corrected by Kevin P. Thompson, Jan 12 2022

a(20)-a(23) from Kevin P. Thompson, Apr 17 2022

a(24) from Kevin P. Thompson, Jun 13 2022