A361968 -id:A361968 - 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

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.

A361971 Record values in A361967.

Original entry on oeis.org

2, 3, 4, 5, 8, 11, 12, 14, 17, 23, 30, 31, 40, 64, 85, 95, 119, 147, 152, 207, 232, 257, 283, 344, 421, 469, 645, 956, 1034, 1306, 1578, 1797, 1943, 2304, 2334, 2877, 3217, 3396, 3536, 3973, 4378, 5171, 5457, 5464, 5659, 7586, 8317, 8430, 10609, 12566, 14469

Offset: 1

Amiram Eldar, Apr 01 2023

nonn

The unitary version of A131934.

Cf. A361966, A361967, A361968.

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

a(n) = A361967(A361968(n)).

a(43)-a(51) from Amiram Eldar, Apr 10 2023

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

Amiram Eldar, Apr 10 2023

nonn

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).

Amiram Eldar, Table of n, a(n) for n = 1..96 (terms below 10^5)

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

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]

A362487 Infinitary highly totient numbers: numbers k that have more solutions x to the equation iphi(x) = k than any smaller k, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

1, 6, 12, 24, 48, 96, 144, 240, 288, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 34560, 51840, 69120, 103680, 120960, 172800, 207360, 241920, 345600, 362880, 414720, 483840, 725760, 967680, 1209600, 1451520, 1935360, 2419200, 2903040, 3628800

Offset: 1

Amiram Eldar, Apr 22 2023

nonn

Indices of records of A362485.

The corresponding numbers of solutions are 2, 4, 6, 10, 14, 18, 22, ... (A362488).

Amiram Eldar, Table of n, a(n) for n = 1..50

Cf. A091732, A362484, A362485, A362486, A362488.

Similar sequences: A097942, A100827, A361968, A362183.

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

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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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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A361971 Record values in A361967.

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

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A362487 Infinitary highly totient numbers: numbers k that have more solutions x to the equation iphi(x) = k than any smaller k, where iphi is the infinitary totient function A091732.

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