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 1-4 of 4 results.

A226117 Numbers n such that phi(sigma(tau(n))) = tau(sigma(phi(n))).

Original entry on oeis.org

1, 3, 4, 5, 14, 17, 20, 21, 22, 26, 51, 63, 65, 66, 72, 76, 80, 84, 90, 100, 106, 112, 132, 135, 150, 152, 165, 182, 190, 196, 221, 222, 232, 246, 255, 290, 291, 292, 294, 320, 326, 375, 386, 396, 424, 450, 460, 489, 530, 561, 567, 585, 588, 600, 606, 608, 615
Offset: 1

Views

Author

Paolo P. Lava, May 27 2013

Keywords

Examples

			For n=23529 we have:
phi(23529)=13200 -> sigma(13200)=46128 -> tau(46128)=30.
tau(23529)=16 -> sigma(16)=31 -> phi(31)=30.

Links

Crossrefs

Cf. A000005, A000010, A000203, A033632, A067160, A076361, A226118, A226119.

Programs

  • Maple
    with(numtheory); A226117:=proc(q) local n;
for n from 1 to q do
if phi(sigma(tau(n)))=tau(sigma(phi(n))) then print(n);
fi; od; end: A226117(10^6);
  • Mathematica
    Select[Range[700],EulerPhi[DivisorSigma[1,DivisorSigma[0,#]]] == DivisorSigma[ 0,DivisorSigma[ 1,EulerPhi[ #]]]&] (* Harvey P. Dale, Dec 12 2021 *)

A226118 Numbers n such that sigma(tau(phi(n))) = phi(tau(sigma(n))).

Original entry on oeis.org

1, 2, 136, 160, 170, 204, 240, 282, 716, 745, 1002, 1077, 1465, 1509, 1578, 1868, 2012, 2157, 2346, 2720, 2760, 3608, 3898, 4101, 4461, 4512, 5066, 5322, 5898, 6189, 7080, 7185, 7341, 7628, 7660, 8108, 8517, 8665, 8698, 8709, 8805, 8922, 8940, 9234, 9745, 9962
Offset: 1

Views

Author

Paolo P. Lava, May 27 2013

Keywords

Examples

			For n=9962 we have:
sigma(9962)=15876 -> tau(15876)=45 -> phi(45)=24.
phi(9962)=4672 -> tau(4672)=14 -> sigma(14)=24.

Links

Crossrefs

Cf. A000005, A000010, A000203, A033632, A067160, A076361, A226117, A226119.

Programs

  • Maple
    with(numtheory); A226118:=proc(q) local n;
for n from 1 to q do
if sigma(tau(phi(n)))=phi(tau(sigma(n))) then print(n);
fi; od; end: A226118(10^6);
  • Mathematica
    Select[Range[10000],EulerPhi[DivisorSigma[0,DivisorSigma[1,#]]] == DivisorSigma[ 1, DivisorSigma[ 0, EulerPhi[#]]]&] (* Harvey P. Dale, May 26 2016 *)

A226119 Numbers such that sigma(phi(tau(n)))=tau(phi(sigma(n))).

Original entry on oeis.org

1, 6, 36, 64, 105, 114, 135, 1980, 2016, 3072, 5120, 7056, 7840, 9216, 16320, 18720, 18900, 23100, 23622, 24003, 25536, 26088, 26733, 28455, 29078, 29337, 29700, 29760, 30597, 30894, 30912, 31155, 31496, 31758, 32361, 33782, 34020, 34286, 36000, 36036, 36099
Offset: 1

Views

Author

Paolo P. Lava, May 27 2013

Keywords

Examples

			29337 is in the sequence since:
sigma(29337)=49152 -> phi(49152)=16384 -> tau(16384)=15.
tau(29337)=16 -> phi(16)=8 -> sigma(8)=15.

Links

Crossrefs

Cf. A000005, A000010, A000203, A033632, A067160, A076361, A226117, A226118.

Programs

  • Maple
    with(numtheory); A226119:=proc(q) local n;
for n from 1 to q do
if sigma(phi(tau(n)))=tau(phi(sigma(n))) then print(n);
fi; od; end: A226119(10^6);
  • Mathematica
    Select[Range[36099], DivisorSigma[1, EulerPhi[DivisorSigma[0, #]]] == DivisorSigma[0, EulerPhi[DivisorSigma[1, #]]] &] (* T. D. Noe, May 28 2013 *)

A291682 Numbers k such that phi(psi(phi(k))) = psi(phi(psi(k))).

Original entry on oeis.org

1, 11, 19, 23, 25, 31, 41, 47, 59, 67, 71, 77, 79, 89, 95, 101, 109, 121, 127, 131, 137, 139, 143, 149, 155, 161, 175, 181, 191, 199, 287, 299, 311, 319, 323, 325, 329, 335, 341, 379, 383, 395, 407, 409, 413, 419, 439, 461, 463, 475, 479, 491, 497, 527, 529, 533, 539, 545, 569, 599, 611, 623, 635
Offset: 1

Views

Author

Altug Alkan, Sep 04 2017

Keywords

Comments

Prime terms are 11, 19, 23, 31, 41, 47, 59, 67, 71, 79, 89, 101, 109, 127, 131, ...
Up to 10^9, twin prime pairs in this sequence are (137, 139), (461, 463), (1019, 1021), (1427, 1429), (2969, 2971), (4229, 4231).

Examples

			11 is a term because phi(psi(phi(11))) = psi(phi(psi(11))).

Links

Crossrefs

Cf. A000010, A001615, A067160, A290002.

Programs

  • Mathematica
    psi[n_] := If[n < 1, 0, n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]]; fQ[n_] := EulerPhi[psi[EulerPhi[n]]] == psi[EulerPhi[psi[n]]]; Select[Range@635, fQ] (* Robert G. Wilson v, Sep 23 2017 *)
  • PARI
    a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = a001615(eulerphi(a001615(n)))==eulerphi(a001615(eulerphi(n))); \\ after Charles R Greathouse IV at A001615
Showing 1-4 of 4 results.

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