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

1,1

Comments

a(70) > 10^10. - Hiroaki Yamanouchi, Sep 28 2015

Links

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..69

Example

sigma*(phi(25)) = sigma*(20) = 24, phi(sigma*(25)) = phi(39) = 24.

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 *)

Crossrefs

Cf. A000203, A066417, A230373, A033632.

Sequence in context: A103853 A192160 A139202 * A125968 A147068 A147012

Adjacent sequences: A243993 A243994 A243995 * A243997 A243998 A243999

Keyword

nonn

Author

Paolo P. Lava, Jun 18 2014

Extensions

a(22)-a(25) from Robert G. Wilson v, Jun 21 2014

a(26)-a(34) from Hiroaki Yamanouchi, Sep 28 2015

Status

approved