A361966 -id:A361966 - cp's OEIS frontend

A362213 Irregular table read by rows in which the n-th row consists of all the numbers m such that cototient(m) = n, where cototient is A051953.

4, 9, 6, 8, 25, 10, 15, 49, 12, 14, 16, 21, 27, 35, 121, 18, 20, 22, 33, 169, 26, 39, 55, 24, 28, 32, 65, 77, 289, 34, 51, 91, 361, 38, 45, 57, 85, 30, 95, 119, 143, 529, 36, 40, 44, 46, 69, 125, 133, 63, 81, 115, 187, 52, 161, 209, 221, 841, 42, 50, 58, 87, 247, 961

Offset: 2

Amiram Eldar, Apr 11 2023

The offset is 2 since cototient(p) = 1 for all primes p.

The 0th row consists of one term, 1, since 1 is the only solution to cototient(x) = 0.

			The table begins:
  n   n-th row
  --  -----------
   2  4;
   3  9;
   4  6, 8;
   5  25;
   6  10;
   7  15, 49;
   8  12, 14, 16;
   9  21, 27;
  10
  11  35, 121;
  12  18, 20, 22;

Amiram Eldar, Table of n, a(n) for n = 2..9674 (rows 2..1000)

Cf. A005278, A051953, A063507, A063740 (row lengths), A063741, A063742, A063748, A100827, A101373, A131825, A131826.

Similar sequences: A032447, A361966, A362180.

With[{max = 50}, cot = Table[n - EulerPhi[n], {n, 1, max^2}]; row[n_] := Position[cot, n] // Flatten; Table[row[n], {n, 2, max}] // Flatten]

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.

A362230 Unitary sparsely totient numbers: numbers k such that m > k implies uphi(m) > uphi(k), where uphi is the unitary totient function (A047994).

Original entry on oeis.org

2, 6, 10, 14, 30, 42, 66, 78, 102, 114, 138, 210, 222, 330, 390, 462, 510, 570, 690, 714, 798, 870, 930, 966, 1110, 1230, 1290, 1302, 1410, 1470, 1590, 1770, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6090, 6510, 6630, 7770, 8610, 9030, 9870, 10230, 11130, 11310

Offset: 1

Amiram Eldar, Apr 12 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..300

The unitary version of A036913.
Record values of A362229.
Cf. A047994, A361966, A362231.

s[n_] := If[(inv = invUPhi[n]) == {}, 0, Max[inv]]; seq[kmax_] := Module[{v = {}, s1, sm = 0}, Do[s1 = s[k]; If[s1 > sm, sm = s1; AppendTo[v, s1]], {k, 1, kmax}]; v]; seq[3000] (* using the function invUPhi from A361966 *)

A362231 a(n) = A047994(A362230(n)).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 20, 24, 32, 36, 44, 48, 72, 80, 96, 120, 128, 144, 176, 192, 216, 224, 240, 264, 288, 320, 336, 360, 368, 384, 416, 464, 480, 576, 768, 864, 960, 1056, 1280, 1344, 1440, 1536, 1728, 1920, 2016, 2208, 2400, 2496, 2688, 2784, 2880, 3168, 3360

Offset: 1

Amiram Eldar, Apr 12 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..300

The unitary version of A036912.
Indices of records of A362229.
Cf. A047994, A361966, A362230.

s[n_] := If[(inv = invUPhi[n]) == {}, 0, Max[inv]]; seq[kmax_] := Module[{v = {}, s1, sm = 0}, Do[s1 = s[k]; If[s1 > sm, sm = s1; AppendTo[v, k]], {k, 1, kmax}]; v]; seq[3000] (* using the function invUPhi from A361966 *)

cp's OEIS Frontend

A362213 Irregular table read by rows in which the n-th row consists of all the numbers m such that cototient(m) = n, where cototient is A051953.

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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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A362230 Unitary sparsely totient numbers: numbers k such that m > k implies uphi(m) > uphi(k), where uphi is the unitary totient function (A047994).

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A362231 a(n) = A047994(A362230(n)).

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