A361971 -id:A361971 - cp's OEIS frontend

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).

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 *)

A361969 Numbers k with a single solution x to the equation uphi(x) = k, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

3, 7, 14, 15, 31, 54, 62, 63, 127, 154, 174, 182, 186, 234, 246, 254, 255, 294, 308, 318, 322, 364, 406, 414, 496, 510, 511, 516, 534, 558, 574, 594, 644, 666, 678, 762, 804, 806, 812, 846, 870, 948, 1022, 1023, 1026, 1036, 1074, 1098, 1146, 1148, 1164, 1204, 1246

Offset: 1

Amiram Eldar, Apr 01 2023

nonn

Numbers k such that A361967(k) = 1.
According to Carmichael's totient function conjecture, there are no numbers with a single solution x to the corresponding equation phi(x) = k, with Euler's totient function (A000010).
A000225(m) = 2^m - 1 is a term for all m >= 2. These are the only odd terms.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Carmichael's totient function conjecture.

Cf. A000010, A000225, A047994, A135347, A361966, A361967, A361968, A361970, A361971.

Select[Range[1250], Length[invUPhi[#]] == 1 &] (* 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.

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

Amiram Eldar, Apr 10 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..96

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

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]

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

A362488 Record values in A362487.

Original entry on oeis.org

2, 4, 6, 10, 14, 18, 22, 30, 34, 40, 48, 58, 60, 92, 136, 146, 184, 232, 240, 342, 478, 518, 638, 772, 830, 924, 1080, 1264, 1330, 1340, 1462, 1824, 2132, 2528, 2710, 3224, 3354, 4084, 4672, 4812, 4976, 5912, 6496, 7606, 8230, 8698, 11472, 12354, 16580, 19250

Offset: 1

Amiram Eldar, Apr 22 2023

nonn

Cf. A362484, A362487.

Similar sequences: A101373, A131934, A361971, A362184.

solnum[n_] := Length[invIPhi[n]]; seq[kmax_] := Module[{s = {}, solmax=0}, Do[sol = solnum[k]; If[sol > solmax, solmax = sol; AppendTo[s, sol]], {k, 1, kmax}]; s]; seq[10^4] (* using the function invIPhi from A362484 *)

a(n) = A362485(A362487(n)).

cp's OEIS Frontend

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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A361969 Numbers k with a single solution x to the equation uphi(x) = 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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A362184 Record values in A362183.

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A362488 Record values in A362487.

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