A347771 -id:A347771 - cp's OEIS frontend

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

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.

A362486 Infinitary nontotient numbers: values not in the range of the infinitary totient function iphi (A091732).

Original entry on oeis.org

5, 7, 9, 11, 13, 14, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 47, 49, 50, 51, 53, 55, 57, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 103, 105, 107, 109, 110, 111

Offset: 1

Amiram Eldar, Apr 22 2023

nonn

Numbers k such that A091732(x) = k has no solution, i.e., A362485(k) = 0.

Most of the odd numbers are in this sequence. Odd numbers that are not here are 1, 3, 15, 45, 255, 765, 3825, 11475, 65535, 196605, 983025, ..., which are the values of iphi at powers of 2.

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

Cf. A091732, A362484, A362485.

Similar sequences: A005277, A005278, A347771, A362182.

Select[Range[120], Length[invIPhi[#]] == 0 &] (* using the function invIPhi from A362484 *)

A362485(a(n)) = 0.

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

Amiram Eldar, Apr 10 2023

nonn

Numbers k such that A362181(k) = 0.

Are 3 and 5 the only odd terms? There are no other odd terms below 10^5.

Amiram Eldar, Table of n, a(n) for n = 1..534 (terms below 10^5)

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

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

A362229 a(n) is the largest m such that uphi(m) = n, where uphi is the unitary totient function (A047994), or a(n) = 0 if no such m exists.

Original entry on oeis.org

2, 6, 4, 10, 0, 14, 8, 30, 0, 22, 0, 42, 0, 24, 16, 34, 0, 38, 0, 66, 0, 46, 0, 78, 0, 54, 0, 58, 0, 62, 32, 102, 0, 0, 0, 114, 0, 0, 0, 110, 0, 86, 0, 138, 0, 94, 0, 210, 0, 0, 0, 106, 0, 76, 0, 174, 0, 118, 0, 186, 0, 96, 64, 170, 0, 134, 0, 0, 0, 142, 0, 222

Offset: 1

Amiram Eldar, Apr 12 2023

nonn

			a(1) = 2 since there are two solutions to uphi(x) = 1: 1 and 2, and 2 is the larger of them.
a(6) = 14 since there are three solutions to uphi(x) = 6: 7, 12 and 14, and 14 is the largest of them.

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

The unitary version of A057635.

Cf. A047994, A347771 (positions of 0's), A361966, A362230 (record values), A362231 (indices of records).

a[n_] := If[(inv = invUPhi[n]) == {}, 0, Max[inv]]; Array[a, 100] (* using the function invUPhi from A361966 *)

a(A347771(n)) = 0.

cp's OEIS Frontend

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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A362486 Infinitary nontotient numbers: values not in the range of the infinitary totient function iphi (A091732).

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

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A362229 a(n) is the largest m such that uphi(m) = n, where uphi is the unitary totient function (A047994), or a(n) = 0 if no such m exists.

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