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A385744 The number of iterations of the infinitary analog of the totient function A384247 that are required to reach from n to 1.

%I A385744 #7 Jul 11 2025 01:27:35
%S A385744 0,1,2,3,4,2,3,4,5,4,5,3,4,3,5,6,7,5,6,4,4,5,6,5,6,4,6,6,7,5,6,7,5,7,
%T A385744 6,6,7,6,6,7,8,4,5,6,8,6,7,6,7,6,8,7,8,6,8,6,7,7,8,6,7,6,7,7,7,5,6,7,
%U A385744 7,6,7,8,9,7,7,7,7,6,7,7,8,8,9,7,8,5,7
%N A385744 The number of iterations of the infinitary analog of the totient function A384247 that are required to reach from n to 1.
%C A385744 First differs from A049865 at n = 24.
%H A385744 Amiram Eldar, <a href="/A385744/b385744.txt">Table of n, a(n) for n = 1..10000</a>
%F A385744 a(n) = a(A384247(n)) + 1 for n >= 2.
%e A385744   n | a(n) | iterations
%e A385744   --+------+----------------------
%e A385744   2 |    1 | 2 -> 1
%e A385744   3 |    2 | 3 -> 2 -> 1
%e A385744   4 |    3 | 4 -> 3 -> 2 -> 1
%e A385744   5 |    4 | 5 -> 4 -> 3 -> 2 -> 1
%e A385744   6 |    2 | 6 -> 2 -> 1
%t A385744 f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); iphi[1] = 1; iphi[n_] := iphi[n] = Times @@ f @@@ FactorInteger[n];
%t A385744 a[n_] := Length @ NestWhileList[iphi, n, # != 1 &] - 1; Array[a, 100]
%o A385744 (PARI) iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2))));}
%o A385744 a(n) = if(n ==  1, 0, 1 + a(iphi(n)));
%Y A385744 Cf. A384247, A385745, A385746, A385747.
%Y A385744 Similar sequences: A003434, A049865, A225320, A333609.
%K A385744 nonn,easy
%O A385744 1,3
%A A385744 _Amiram Eldar_, Jul 08 2025