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 1-7 of 7 results.

A362485 Number of numbers k such that iphi(k) = n, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

2, 2, 2, 2, 0, 4, 0, 4, 0, 2, 0, 6, 0, 0, 2, 4, 0, 4, 0, 2, 0, 2, 0, 10, 0, 0, 0, 2, 0, 6, 0, 4, 0, 0, 0, 8, 0, 0, 0, 4, 0, 2, 0, 2, 2, 2, 0, 14, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 10, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 14, 0, 0, 0, 0, 0, 2, 0, 8, 0, 2, 0, 4, 0, 0
Offset: 1

Views

Author

Amiram Eldar, Apr 22 2023

Keywords

Comments

a(n) is even for all n, because if k is a solution to iphi(k) = n, and A007814(k) is even, then 2*k is also a solution, i.e., iphi(2*k) = n.

Links

Crossrefs

Row lengths of A362484.
Cf. A007814, A091732, A362486 (positions of 0's), A362487 (indices of records).
Similar sequences: A014197, A063740, A361967, A362181.

Programs

  • Mathematica
    a[n_] := Length[invIPhi[n]]; Array[a, 100] (* using the function invIPhi from A362484 *)

Formula

a(A362486(n)) = 0.

A362180 Irregular table read by rows in which the n-th row consists of all the numbers m such that A323410(m) = n.

Original entry on oeis.org

6, 10, 12, 15, 14, 20, 21, 18, 24, 28, 35, 22, 36, 40, 33, 45, 26, 44, 56, 39, 55, 63, 52, 72, 65, 77, 34, 48, 88, 51, 91, 99, 38, 68, 80, 104, 57, 85, 117, 30, 76, 112, 95, 119, 143, 46, 136, 144, 69, 133, 153, 50, 92, 152, 176, 75, 115, 171, 187, 54, 100, 208
Offset: 2

Views

Author

Amiram Eldar, Apr 10 2023

Keywords

Comments

The offset is 2 since A323410(p) = 1 for all prime powers p (A246655).
The 0th row consists of one term, 1, since 1 is the only solution to A323410(x) = 0.

Examples

			The table begins:
  n   n-th row
  --  -----------
   2
   3
   4  6;
   5
   6  10, 12;
   7  15;
   8  14, 20;
   9  21;
  10  18, 24, 28;
  11  35;
  12  22, 36, 40;

Links

Crossrefs

Cf. A246655, A323410, A362181 (row lengths).
Similar sequences: A032447, A361966, A362213.

Programs

  • Mathematica
    ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0;
With[{max = 28}, ucot = Table[ucototient[n], {n, 1, max^2}]; row[n_] := Position[ucot, n] // Flatten; Table[row[n], {n, 2, max}] // Flatten]

A362183 Unitary highly cototient numbers: numbers k that have more solutions x to the equation A323410(x) = k than any smaller k.

Original entry on oeis.org

0, 6, 10, 20, 31, 47, 53, 65, 77, 89, 113, 119, 149, 167, 179, 209, 293, 299, 329, 359, 389, 419, 479, 509, 599, 629, 779, 839, 989, 1049, 1139, 1259, 1469, 1559, 1649, 1679, 1889, 2099, 2309, 2729, 3149, 3359, 3569, 3989, 4289, 4409, 4619, 5249, 5459, 6089, 6509
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2023

Keywords

Comments

Indices of records of A362181.
The corresponding numbers of solutions are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 21, ... (A362184).

Links

Crossrefs

Cf. A323410, A362181, A362184.
The unitary version of A100827.
Similar sequences: A097942, A361968.

Programs

  • Mathematica
    ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 300}, solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, solnum[[i]]++], {k, 2, max^2}]; s = {0}; solmax=1; Do[sol = solnum[[k]]; If[sol > solmax, solmax = sol; AppendTo[s, k]], {k, 2, max}]; s]

A362184 Record values in A362183.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 21, 23, 25, 26, 27, 31, 33, 34, 37, 38, 45, 49, 54, 59, 62, 64, 71, 80, 81, 84, 92, 99, 106, 122, 137, 145, 147, 167, 174, 180, 183, 203, 211, 231, 232, 251, 253, 283, 289, 306, 318, 342, 362, 378, 410, 412, 453
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2023

Keywords

Links

Crossrefs

The unitary version of A101373.
Cf. A362181, A362183.
Similar sequences: A131934, A361971.

Programs

  • Mathematica
    ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 300}, solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, solnum[[i]]++], {k, 2, max^2}]; s = {1}; solmax=1; Do[sol = solnum[[k]]; If[sol > solmax, solmax = sol; AppendTo[s, sol]], {k, 2, max}]; s]

Formula

a(n) = A362181(A362183(n)).

A362185 Numbers k with a single solution x to the equation A323410(x) = k.

Original entry on oeis.org

0, 4, 7, 9, 11, 216, 218, 220, 546, 652, 666, 700, 834, 850, 906, 924, 996, 1242, 1386, 1476, 1506, 1516, 1596, 1646, 1662, 1758, 1770, 1858, 1890, 1900, 1946, 2046, 2170, 2262, 2352, 2422, 2578, 2626, 2668, 2682, 2814, 2842, 2980, 2992, 3010, 3048, 3100, 3154
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2023

Keywords

Comments

Numbers k such that A362181(k) = 1.

Examples

			0 is a term since there is only one solution, x = 1, to A323410(x) = 0.

Links

Crossrefs

Cf. A323410, A362181.
The unitary version of A131825.
Similar sequence: A361969.

Programs

  • Mathematica
    ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 300}, solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, solnum[[i]]++], {k, 2, max^2}]; Join[{0}, Position[solnum, 1] // Flatten]]

A362182 Unitary noncototient numbers: numbers k such that A323410(x) = k has no solution.

Original entry on oeis.org

2, 3, 5, 330, 1206, 1210, 1656, 1718, 1806, 1866, 1926, 2376, 2982, 3162, 3186, 3342, 4012, 4062, 4194, 4326, 4502, 4662, 4810, 5322, 5466, 6172, 6402, 6462, 6534, 6546, 6672, 6756, 7266, 7430, 7866, 8030, 8140, 8286, 8386, 8562, 8586, 8860, 9114, 9370, 9516, 9906
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2023

Keywords

Comments

Numbers k such that A362181(k) = 0.
Are 3 and 5 the only odd terms? There are no other odd terms below 10^5.

Links

Crossrefs

Cf. A323410, A362181.
The unitary version of A005278.
Similar sequences: A007617, A347771.

Programs

  • Mathematica
    ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 2000}, Complement[Range[max], Table[ucototient[n], {n, 1, max^2}]]]

A362186 a(n) is the least number k such that the equation A323410(x) = k has exactly n solutions, or -1 if no such k exists.

Original entry on oeis.org

2, 0, 6, 10, 20, 31, 47, 53, 65, 77, 89, 113, 125, 119, 149, 173, 167, 179, 233, 279, 239, 209, 439, 293, 365, 299, 329, 359, 455, 521, 467, 389, 461, 419, 479, 773, 539, 509, 599, 845, 671, 791, 749, 719, 659, 629, 809, 1055, 881, 779, 899, 965, 929, 1121, 839, 1403
Offset: 0

Views

Author

Amiram Eldar, Apr 10 2023

Keywords

Comments

Is there any n for which a(n) = -1?

Links

Crossrefs

The unitary version of A063507.
Cf. A323410, A362181.
Similar sequences: A007374, A361970.

Programs

  • Mathematica
    ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 300}, solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, solnum[[i]]++], {k, 2, max^2}]; Join[{2, 0}, TakeWhile[FirstPosition[ solnum, #] & /@ Range[2, max] // Flatten, NumberQ]]]

Formula

A362181(a(n)) = n.
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.