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 1-6 of 6 results.

A330273 Infinitary perfect totient numbers: numbers that equal to the sum of their iterated infinitary totient function (A091732).

Original entry on oeis.org

3, 10, 21, 44, 93, 118, 170, 320, 548, 3596, 3620, 4772, 5564, 18260, 33051, 256425, 403700, 1071129, 1790160, 2318180, 3968852, 4027375, 10001319, 11270012, 12048740, 13358121, 31741593, 46271673, 56149161, 4344134553
Offset: 1

Views

Author

Amiram Eldar, Dec 13 2019

Keywords

Comments

The infinitary version of A082897 (perfect totient numbers), in which the infinitary totient function iphi (A091732) replaces the Euler totient function (A000010).

Examples

			10 is an infinitary perfect totient number because iphi(10) + iphi(iphi(10)) + ... = 4 + 3 + 2 + 1 = 10.

Crossrefs

Cf. A082897, A091732, A286067.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); iphi[1] = 1; iphi[n] := iphi[n] = Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); infPerfTotQ[n_] := Plus @@ FixedPointList[iphi@# &, n] == 2 n + 1; Select[Range[1000], infPerfTotQ]

A329153 Sum of the iterated unitary totient function (A047994).

Original entry on oeis.org

0, 1, 3, 6, 10, 3, 9, 16, 24, 10, 20, 9, 21, 9, 24, 39, 55, 24, 42, 21, 21, 20, 42, 23, 47, 21, 47, 42, 70, 24, 54, 85, 41, 55, 47, 47, 83, 42, 47, 70, 110, 21, 63, 54, 117, 42, 88, 54, 102, 47, 117, 83, 135, 47, 110, 63, 83, 70, 128, 47, 107, 54, 102, 165, 102
Offset: 1

Views

Author

Amiram Eldar, Feb 25 2020

Keywords

Comments

Analogous to A092693 with the unitary totient function uphi instead of the Euler totient function phi (A000010).

Examples

			a(4) = uphi(4) + uphi(uphi(4)) + uphi(uphi(uphi(4))) = 3 + 2 + 1 = 6.

Links

Crossrefs

Cf. A003271, A047994, A049865, A092693, A225172, A225173, A286067.

Programs

  • Mathematica
    uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); Table[Plus @@ FixedPointList[uphi, n] - n - 1, {n, 1, 100}]

Formula

a(n) = n for n in A286067.

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;

A333103 Quasiperfect cototient numbers: numbers k such that the sum of the iterated cototient function of k is equal to k+1.

Original entry on oeis.org

6, 14, 62, 254, 16382, 78585, 87465, 262142, 1048574
Offset: 1

Views

Author

Amiram Eldar, Mar 07 2020

Keywords

Comments

If m is in A050475 then 2^m - 2 is a term.
3*10^8 < a(10) <= 4294967292.

Examples

			6 is a term since A051953(6) = 4, A051953(4) = 2, A051953(2) = 1, and 4 + 2 + 1 = 7 = 6 + 1.

Links

Crossrefs

Cf. A050475, A051953, A082897, A286067, A330273.

Programs

  • Mathematica
    cot[n_] := n - EulerPhi[n]; s[n_] := Plus @@ FixedPointList[cot, n]; Select[Range[10^5], s[#] == 2*# + 1 &]

A333873 Numbers that equal to the sum of their iterated absolute Möbius divisor function (A173557).

Original entry on oeis.org

3, 5, 17, 257, 413, 611, 1391, 1589, 1903, 2327, 5599, 27959, 29623, 36647, 36983, 38863, 42851, 43919, 46463, 49513, 65537, 76759, 82969, 86567, 88759, 96839, 111179, 116479, 129307, 171191, 184979, 213041, 277619, 301157, 310519, 346151, 362263, 372227, 375167
Offset: 1

Views

Author

Amiram Eldar, Apr 08 2020

Keywords

Comments

A variant of A082897 (perfect totient numbers) in which the absolute Möbius divisor function (A173557) replaces the Euler totient function (A000010).

Examples

			5 is a term since A173557(5) = 4, A173557(4) = 1, and 4 + 1 = 5.

Links

Crossrefs

A019434 is a subsequence.
Cf. A082897, A173557, A286067, A330273, A333871.

Programs

  • Mathematica
    f[p_, e_] := p - 1; u[1] = 1; u[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^4], Plus @@ FixedPointList[u, #] == 2*# + 1 &]

A333104 Unitary quasiperfect cototient numbers: numbers k such that the sum of the iterated unitary cototient function of k is equal to k+1.

Original entry on oeis.org

10, 22, 98, 118, 230, 266, 1452, 88894, 114214, 1274198, 51675986, 61177358, 82986118
Offset: 1

Views

Author

Amiram Eldar, Mar 07 2020

Keywords

Comments

a(14) > 10^9.

Examples

			10 is a term since A323410(10) = 6, A323410(6) = 4, A323410(4) = 1 and 6 + 4 + 1 = 11 = 10 + 1.

Crossrefs

Cf. A082897, A286067, A323410, A330273, A333103.

Programs

  • Mathematica
    uphi[0] = 0; uphi[1] = 1; uphi[n_] := (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]; ucot[n_] := n - uphi[n]; Select[Range[10^4], Plus @@ FixedPointList[ucot, #] == 2*# + 1 &]
Showing 1-6 of 6 results.

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