A243996 Numbers n such that phi(sigma*(n)) = sigma*(phi(n)), where sigma*(n) is the sum of anti-divisors of n and phi(n) is the Euler totient function.
7, 9, 20, 25, 80, 143, 825, 3117, 3216, 22774, 52026, 55804, 138276, 187733, 228384, 265545, 320766, 549540, 830814, 839784, 901376, 1293552, 1315776, 2635866, 6771114, 11126800, 12087848, 24351460, 49382242, 52344292, 60063744, 65980038, 78279016, 97638080
Offset: 1
Keywords
Examples
sigma*(phi(25)) = sigma*(20) = 24, phi(sigma*(25)) = phi(39) = 24.
Links
- Hiroaki Yamanouchi, Table of n, a(n) for n = 1..69
Programs
-
Maple
with(numtheory): P:=proc(q) local a,b,c,d,j,k,n; for n from 1 to q do k:=0; j:=n; while j mod 2<>1 do k:=k+1; j:=j/2; od; a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2; k:=0; c:=phi(n); j:=phi(n); while j mod 2<>1 do k:=k+1; j:=j/2; od; b:=sigma(2*c+1)+sigma(2*c-1)+sigma(c/2^k)*2^(k+1)-6*c-2; if b=phi(a) then print(n); fi; od; end: P(10^10);
-
Mathematica
antiDivisors[n_] := Select[ Union[ Join[ Select[ Divisors[2 n - 1], OddQ[#] && # != 1 &], Select[ Divisors[ 2n + 1], OddQ[#] && # != 1 &], 2n/Select[ Divisors[ 2n], OddQ[#] && # != 1 &]]], # < n &]; fQ[n_] := EulerPhi@ Total@ antiDivisors@ n == Total@ antiDivisors@ EulerPhi@ n; k = 3; lst = {}; While[k < 10000001, If[ fQ@ k, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Jun 21 2014 *)
Extensions
a(22)-a(25) from Robert G. Wilson v, Jun 21 2014
a(26)-a(34) from Hiroaki Yamanouchi, Sep 28 2015
Comments