A385746 Numbers that are equal to the sum of their iterated infinitary analog of the totient function A384247.

3, 10, 18, 21, 48, 160, 288, 3252, 9304, 13965, 68526, 719631, 1531101, 1954782, 28900572, 39189195, 14708055957

Offset: 1

Author

Amiram Eldar, Jul 08 2025

Keywords

• nonn
• more

Comments

Numbers k such that A385745(k) = k.

Examples

			  n | a(n) | iterations                        | sum
  --+------+-----------------------------------+----------------------------
  1 |    3 | 3 -> 2 -> 1                       | 2 + 1 = 3
  2 |   10 | 10 -> 4 -> 3 -> 2 -> 1            | 4 + 3 + 2 + 1 = 10
  3 |   18 | 18 -> 8 -> 4 -> 3 -> 2 -> 1       | 8 + 4 + 3 + 2 + 1 = 18
  4 |   21 | 21 -> 12 -> 6 -> 2 -> 1           | 12 + 6 + 2 + 1 = 21
  5 |   48 | 48 -> 30 -> 8 -> 4 -> 3 -> 2 -> 1 | 30 + 8 + 4 + 3 + 2 + 1 = 48
		

Crossrefs

Cf. A384247, A385744, A385745, A385747.

Similar sequences: A082897, A286067, A330273.

Programs

Mathematica

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); iphi[1] = 1; iphi[n_] := iphi[n] = Times @@ f @@@ FactorInteger[n];
infPerfTotQ[n_] := Plus @@ FixedPointList[iphi, n] == 2*n + 1; infPerfTotQ[1] = False; Select[Range[10^5], infPerfTotQ]

PARI

iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2)))); }
s(n) = if(n == 1, 0, my(i = iphi(n)); i + s(i));
isok(k) = s(k) == k;