A385745 The sum of the iterated infinitary analog of the totient function A384247 when started at n.

0, 1, 3, 6, 10, 3, 9, 10, 18, 10, 20, 9, 21, 9, 18, 33, 49, 18, 36, 21, 21, 20, 42, 18, 42, 21, 36, 36, 64, 18, 48, 49, 41, 49, 42, 42, 78, 36, 42, 49, 89, 21, 63, 48, 81, 42, 88, 48, 96, 42, 81, 78, 130, 36, 89, 42, 78, 64, 122, 42, 102, 48, 96, 96, 96, 41, 107

Offset

1,3

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

  n | iterations            | a(n)
  --+-----------------------+--------------------
  2 | 2 -> 1                | 1
  3 | 3 -> 2 -> 1           | 2 + 1 = 3
  4 | 4 -> 3 -> 2 -> 1      | 3 + 2 + 1 = 6
  5 | 5 -> 4 -> 3 -> 2 -> 1 | 4 + 3 + 2 + 1 = 10
  6 | 6 -> 2 -> 1           | 2 + 1 = 3

Mathematica

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); iphi[1] = 1; iphi[n_] := iphi[n] = Times @@ f @@@ FactorInteger[n];
a[n_] := Plus @@ NestWhileList[iphi, n, # != 1 &] - n; Array[a, 100]

Prog

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

Crossrefs

Cf. A384247, A385744, A385746, A385747.

Similar sequences: A092693, A329153, A333611.

Sequence in context: A330376 A120028 A333611 * A329153 A337771 A232175

Adjacent sequences: A385742 A385743 A385744 * A385746 A385747 A385748

Keyword

nonn,easy

Author

Amiram Eldar, Jul 08 2025

Status

approved