cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Amiram Eldar, Jul 08 2025

Keywords

Comments

First differs from A049865 at n = 24.

Examples

			  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

Links

Crossrefs

Cf. A384247, A385745, A385746, A385747.
Similar sequences: A003434, A049865, A225320, A333609.

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_] := Length @ NestWhileList[iphi, n, # != 1 &] - 1; 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, 1 + a(iphi(n)));

Formula

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

A385746 Numbers that are equal to the sum of their iterated infinitary analog of the totient function A384247.

Original entry on oeis.org

3, 10, 18, 21, 48, 160, 288, 3252, 9304, 13965, 68526, 719631, 1531101, 1954782, 28900572, 39189195, 14708055957
Offset: 1

Views

Author

Amiram Eldar, Jul 08 2025

Keywords

Comments

Numbers k such that A385745(k) = k.

Examples

			  n | a(n) | iterations                        | sum
  --+------+-----------------------------------+----------------------------
  1 |    3 | 3 -> 2 -> 1                       | 2 + 1 = 3
  2 |   10 | 10 -> 4 -> 3 -> 2 -> 1            | 4 + 3 + 2 + 1 = 10
  3 |   18 | 18 -> 8 -> 4 -> 3 -> 2 -> 1       | 8 + 4 + 3 + 2 + 1 = 18
  4 |   21 | 21 -> 12 -> 6 -> 2 -> 1           | 12 + 6 + 2 + 1 = 21
  5 |   48 | 48 -> 30 -> 8 -> 4 -> 3 -> 2 -> 1 | 30 + 8 + 4 + 3 + 2 + 1 = 48

Crossrefs

Cf. A384247, A385744, A385745, A385747.
Similar sequences: A082897, A286067, A330273.

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];
infPerfTotQ[n_] := Plus @@ FixedPointList[iphi, n] == 2*n + 1; infPerfTotQ[1] = False; Select[Range[10^5], infPerfTotQ]
  • PARI
    iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2)))); }
s(n) = if(n == 1, 0, my(i = iphi(n)); i + s(i));
isok(k) = s(k) == k;

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

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 08 2025

Keywords

Comments

a(n) is the least number k such that A385744(k) = n.
Also, indices of records of A385744.

Examples

			  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

Crossrefs

Cf. A384247, A385744, A385745, A385746.
Similar sequences: A003271, A005424, A007755, A333610.

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];
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]
  • PARI
    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++));}
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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