A333610 -id:A333610 - cp's OEIS frontend

A385747 Least number that reaches 1 after exactly n iterations of the infinitary analog of the totient function A384247.

1, 2, 3, 4, 5, 9, 16, 17, 41, 73, 101, 197, 467, 829, 1109, 2761, 4849, 7831, 12401, 26189, 52379, 85853, 139589, 237007, 395533, 947043, 1967027, 3446033, 5396427, 9510437, 17502533, 35005067, 71202449, 90187609, 164664701, 395199461, 705113873, 1265735729, 1803553457

Offset: 0

Amiram Eldar, Jul 08 2025

nonn

a(n) is the least number k such that A385744(k) = n.

Also, indices of records of A385744.

			  n | a(n) | iterations
  --+------+---------------------------
  1 |    2 | 2 -> 1
  2 |    3 | 3 -> 2 -> 1
  3 |    4 | 4 -> 3 -> 2 -> 1
  4 |    5 | 5 -> 4 -> 3 -> 2 -> 1
  5 |    9 | 9 -> 8 -> 4 -> 3 -> 2 -> 1

Cf. A384247, A385744, A385745, A385746.

Similar sequences: A003271, A005424, A007755, A333610.

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); iphi[1] = 1; iphi[n_] := iphi[n] = Times @@ f @@@ FactorInteger[n];
numiter[n_] := Length @ NestWhileList[iphi, n, # != 1 &] - 1;
seq[len_] := Module[{s = {}, k = 0, i = 0}, While[Length[s] < len, k++; If[numiter[k] == i, AppendTo[s, k]; i++]]; s]; seq[25]

iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2))));}
numiter(n) = if(n ==  1, 0, 1 + numiter(iphi(n)));
list(len) = {my(k = 0, i = 0, c = 0); while(c < len, k++; if(numiter(k) == i, c++; print1(k, ", "); i++));}

A362025 a(n) is the least number that reaches 1 after n iterations of the infinitary totient function A064380.

Original entry on oeis.org

2, 3, 4, 5, 9, 11, 16, 17, 28, 29, 46, 47, 99, 145, 167, 205, 314, 397, 437, 793, 851, 1137, 1693, 2453, 2771, 2989, 3701, 5099, 6801, 9299, 12031, 15811, 16816, 21520, 21521, 29547, 39685, 62077, 83191, 103473, 112117, 149535, 157159, 196049, 200267, 303022

Offset: 1

Amiram Eldar, Apr 05 2023

nonn

Cf. A064380.
Indices of records of A362024.
Similar sequences: A003271, A007755, A333610.

infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]];
iphi[n_] := Sum[Boole[infCoprimeQ[j, n]], {j, 1, n - 1}];
numiter[n_] := Length@ NestWhileList[iphi, n, # > 1 &] - 1;
seq[kmax_] := Module[{v = {}, n = 1}, Do[If[numiter[k] == n, AppendTo[v, k]; n++], {k, 2, kmax}]; v]; seq[1000]

isinfcoprime(n1, n2) = {my(g = gcd(n1, n2), p, e1, e2); if(g == 1, return(1)); p = factor(g)[, 1]; for(i=1, #p, e1 = valuation(n1, p[i]); e2 = valuation(n2, p[i]); if(bitand(e1, e2) > 0, return(0))); 1; }
iphi(n) = sum(j = 1, n-1, isinfcoprime(j, n));
numiter(n) = if(n==2, 1, numiter(iphi(n)) + 1);
lista(kmax) = {my(n = 1); for(k = 2, kmax, if(numiter(k) == n, print1(k, ", "); n++)); }

A362024(a(n)) = n, and A362024(k) < n for all k < a(n).

cp's OEIS Frontend

A385747 Least number that reaches 1 after exactly n iterations of the infinitary analog of the totient function A384247.

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