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

Author

Amiram Eldar, Jul 08 2025

Keywords

• nonn
• easy

Examples

			  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
		

Links

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

Crossrefs

Cf. A384247, A385744, A385746, A385747.

Similar sequences: A092693, A329153, A333611.

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];
a[n_] := Plus @@ NestWhileList[iphi, n, # != 1 &] - n; Array[a, 100]

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));