A361969 -id:A361969 - cp's OEIS frontend

A362212 a(n) is the unique solution to A047994(x) = A361969(n).

4, 8, 24, 16, 32, 76, 96, 64, 128, 184, 236, 216, 224, 316, 332, 384, 256, 344, 552, 428, 376, 424, 472, 556, 544, 768, 512, 692, 716, 608, 664, 796, 1128, 892, 908, 896, 1076, 864, 1416, 1132, 944, 1268, 1536, 1024, 1372, 1192, 1436, 1468, 1532, 1992, 1556, 1384

Offset: 1

Amiram Eldar, Apr 11 2023

nonn

Are all the terms divisible by 4?

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A047994, A135347, A361966, A361969.

Similar sequences: A131826, A362211.

invUPhi[#][[1]]& /@ Select[Range[1250], Length[invUPhi[#]] == 1 &] (* using the function invUPhi from A361966 *)

a(n) = A135347(A361969(n)).

A047994(a(n)) = A361969(n).

A361966 Irregular table read by rows in which the n-th row consists of all the numbers m such that uphi(m) = n, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

1, 2, 3, 6, 4, 5, 10, 7, 12, 14, 8, 9, 15, 18, 30, 11, 22, 13, 20, 21, 26, 42, 24, 16, 17, 34, 19, 28, 38, 33, 66, 23, 46, 25, 35, 36, 39, 50, 60, 70, 78, 27, 54, 29, 40, 58, 31, 44, 48, 62, 32, 45, 51, 90, 102, 37, 52, 57, 74, 84, 114, 41, 55, 82, 110, 43, 56, 86

Offset: 1

Amiram Eldar, Apr 01 2023

			The table begins:
  n   n-th row
  --  --------
   1  1, 2;
   2  3, 6;
   3  4;
   4  5, 10;
   5
   6  7, 12, 14;
   7  8;
   8  9, 15, 18, 30;
   9
  10  11, 22;
  11
  12  13, 20, 21, 26, 42;

Amiram Eldar, Table of n, a(n) for n = 1..16006 (first 10000 rows)

The unitary version of A032447.

Cf. A047994, A135347, A162247, A361967 (row lengths), A361968, A361969, A361970, A361971.

invUPhi[n_] := Module[{fct = f[n], sol}, sol = Times @@@ (1 + Select[fct, UnsameQ @@ # && (Length[#] == 1 || CoprimeQ @@ (# + 1)) && Times @@ PrimeNu[# + 1] == 1 &]); Sort@ Join[sol, 2*Select[sol, OddQ]]]; invUPhi[1] = {1, 2}; Table[invUPhi[n], {n, 1, 50}] // Flatten (* using the function f by T. D. Noe at A162247 *)

A361967 Number of numbers k such that uphi(k) = n, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

2, 2, 1, 2, 0, 3, 1, 4, 0, 2, 0, 5, 0, 1, 1, 2, 0, 3, 0, 2, 0, 2, 0, 8, 0, 2, 0, 3, 0, 4, 1, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 3, 0, 2, 0, 2, 0, 11, 0, 0, 0, 2, 0, 1, 0, 4, 0, 2, 0, 8, 0, 1, 1, 2, 0, 3, 0, 0, 0, 3, 0, 11, 0, 0, 0, 0, 0, 3, 0, 8, 0, 2, 0, 5, 0, 0, 0

Offset: 1

Amiram Eldar, Apr 01 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Row lengths of A361966.
The unitary version of A014197.
Cf. A047994, A135347, A327837, A347771 (positions of 0's), A361966, A361968 (indices of records), A361969 (positions of 1's), A361970, A361971 (record values).

a[n_] := Length[invUPhi[n]]; Array[a, 100] (* using the function invUPhi from A361966 *)

a(A347771(n)) = 0.
a(A361969(n)) = 1.
a(A361970(n)) = n.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A327837. - Amiram Eldar, Dec 24 2024

A361968 Unitary highly totient numbers: numbers k that have more solutions x to the equation uphi(x) = k than any smaller k, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

1, 6, 8, 12, 24, 48, 96, 120, 144, 240, 480, 576, 720, 1440, 2880, 4320, 5760, 8640, 10080, 17280, 20160, 30240, 34560, 40320, 60480, 80640, 120960, 241920, 362880, 483840, 725760, 967680, 1209600, 1451520, 2177280, 2419200, 2903040, 3628800, 4354560, 4838400

Offset: 1

Amiram Eldar, Apr 01 2023

nonn

Indices of records of A361967.

The corresponding numbers of solutions are 2, 3, 4, 5, 8, 11, ... (A361971).

Amiram Eldar, Table of n, a(n) for n = 1..51

The unitary version of A097942.

Cf. A047994, A135347, A347771, A361966, A361967, A361969, A361970, A361971.

solnum[n_] :=  Length[invUPhi[n]]; seq[kmax_] := Module[{s = {}, solmax=0}, Do[sol = solnum[k]; If[sol > solmax, solmax = sol; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^5] (* using the function invUPhi from A361966 *)

A361970 a(n) is the least number k such that the equation uphi(x) = k has exactly n solutions, or -1 if no such k exists, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

5, 1, 2, 6, 8, 12, 36, 156, 24, 552, 168, 48, 96, 420, 120, 192, 3264, 144, 384, 336, 1536, 288, 360, 240, 672, 1200, 3888, 1080, 4896, 1584, 480, 576, 7056, 4992, 864, 1872, 1152, 3120, 960, 2400, 720, 2520, 30960, 2688, 19968, 1680, 1728, 1920, 2016, 2304, 12000

Offset: 0

Amiram Eldar, Apr 01 2023

nonn

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

Amiram Eldar, Table of n, a(n) for n = 0..300

The unitary version of A007374.

Cf. A047994, A135347, A347771, A361966, A361967, A361968, A361969, A361971.

solnum[n_] :=  Length[invUPhi[n]]; seq[len_, kmax_] := Module[{s = Table[-1, {len}], c = 0, k = 1, ind}, While[k < kmax && c < len, ind = solnum[k] + 1; If[ind <= len && s[[ind]] < 0, c++; s[[ind]] = k]; k++]; s]; seq[50, 10^5] (* using the function invUPhi from A361966 *)

A361967(a(n)) = 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

Amiram Eldar, Apr 10 2023

nonn

Numbers k such that A362181(k) = 1.

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

Amiram Eldar, Table of n, a(n) for n = 1..2335 (terms below 10^5)

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

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]]

A362664 Numbers k with exactly two solutions x to the equation iphi(x) = k, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

1, 2, 3, 4, 10, 15, 20, 22, 28, 42, 44, 45, 46, 52, 54, 56, 58, 70, 78, 82, 92, 100, 102, 104, 106, 116, 130, 136, 140, 148, 162, 164, 166, 172, 174, 178, 184, 190, 196, 200, 204, 208, 212, 220, 222, 226, 228, 234, 238, 246, 250, 255, 260, 262, 268, 272, 282, 292, 296

Offset: 1

Amiram Eldar, Apr 29 2023

nonn

Numbers k such that A362485(k) = 2.

There are no numbers k with a single solution to iphi(x) = k, because if iphi(x) = k, and A007814(x) is even, then 2*x is also a solution, i.e., iphi(2*x) = k.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007814, A091732, A362484, A362485.

Similar sequences: A361969, A362185.

Select[Range[300], Length[invIPhi[#]] == 2 &] (* using the function invIPhi from A362484 *)

cp's OEIS Frontend

A362212 a(n) is the unique solution to A047994(x) = A361969(n).

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A361966 Irregular table read by rows in which the n-th row consists of all the numbers m such that uphi(m) = n, where uphi is the unitary totient function (A047994).

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A361967 Number of numbers k such that uphi(k) = n, where uphi is the unitary totient function (A047994).

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A361968 Unitary highly totient numbers: numbers k that have more solutions x to the equation uphi(x) = k than any smaller k, where uphi is the unitary totient function (A047994).

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A361970 a(n) is the least number k such that the equation uphi(x) = k has exactly n solutions, or -1 if no such k exists, where uphi is the unitary totient function (A047994).

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

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A362664 Numbers k with exactly two solutions x to the equation iphi(x) = k, where iphi is the infinitary totient function A091732.

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