A225320 -id:A225320 - cp's OEIS frontend

A005424 Smallest number that requires n iterations of the bi-unitary totient function (A116550) to reach 1.

2, 3, 4, 5, 8, 9, 13, 16, 17, 24, 25, 35, 44, 63, 64, 91, 97, 128, 193, 221, 259, 324, 353, 391, 477, 702, 929, 1188, 1269, 1589, 1613, 2017, 2309, 2623, 3397, 4064, 4781, 5468, 6515, 6887, 9213, 12286, 12887, 14009, 16564, 16897, 17803, 30428, 36256

Offset: 1

N. J. A. Sloane

Let p(n) = number of unitary divisors k of n, k

M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..97 (terms 1..78 from Donovan Johnson)

Cf. A116550, A225175, A225176.

L := [seq(0,i=0..100)] ;
for n from 1 do
    itr := A225320(n) ;
    if itr < nops(L) then
        if op(itr,L) = 0 then
            L := subsop(itr=n,L) ;
            print(L) ;
        end if;
    end if;
end do: # R. J. Mathar, May 02 2013

A116550[1] = 1; A116550[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; A225320[n_] := A225320[n] = If[n == 1, 0, 1+A225320[A116550[n]]]; L = Array[0&, 100]; For[n = 1, n <= 40000, n++, itr = A225320[n]; If[itr < Length[L], If[L[[itr]] == 0, L = ReplacePart[L, itr -> n]; Print[Select[L, Positive] // Last]]]]; Select[L, Positive] (* Jean-François Alcover, Jan 13 2014, after R. J. Mathar *)

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

Original entry on oeis.org

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, 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, 7, 6, 7, 8, 9, 7, 7, 7, 7, 6, 7, 7, 8, 8, 9, 7, 8, 5, 7

Offset: 1

Amiram Eldar, Jul 08 2025

First differs from A049865 at n = 24.

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

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

Cf. A384247, A385745, A385746, A385747.

Similar sequences: A003434, A049865, A225320, A333609.

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

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, 1 + a(iphi(n)));

a(n) = a(A384247(n)) + 1 for n >= 2.

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A005424 Smallest number that requires n iterations of the bi-unitary totient function (A116550) to reach 1.

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