A362484 -id:A362484 - cp's OEIS frontend

A362485 Number of numbers k such that iphi(k) = n, where iphi is the infinitary totient function A091732.

2, 2, 2, 2, 0, 4, 0, 4, 0, 2, 0, 6, 0, 0, 2, 4, 0, 4, 0, 2, 0, 2, 0, 10, 0, 0, 0, 2, 0, 6, 0, 4, 0, 0, 0, 8, 0, 0, 0, 4, 0, 2, 0, 2, 2, 2, 0, 14, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 10, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 14, 0, 0, 0, 0, 0, 2, 0, 8, 0, 2, 0, 4, 0, 0

Offset: 1

Amiram Eldar, Apr 22 2023

nonn

a(n) is even for all n, because if k is a solution to iphi(k) = n, and A007814(k) is even, then 2*k is also a solution, i.e., iphi(2*k) = n.

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

Row lengths of A362484.
Cf. A007814, A091732, A362486 (positions of 0's), A362487 (indices of records).
Similar sequences: A014197, A063740, A361967, A362181.

a[n_] := Length[invIPhi[n]]; Array[a, 100] (* using the function invIPhi from A362484 *)

a(A362486(n)) = 0.

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.

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 *)

A362666 a(n) is the largest m such that iphi(m) = n, where iphi is the infinitary totient function A091732, or a(n) = 0 if no such m exists.

Original entry on oeis.org

2, 6, 8, 10, 0, 24, 0, 30, 0, 22, 0, 42, 0, 0, 32, 54, 0, 56, 0, 66, 0, 46, 0, 120, 0, 0, 0, 58, 0, 96, 0, 102, 0, 0, 0, 168, 0, 0, 0, 110, 0, 86, 0, 138, 128, 94, 0, 216, 0, 0, 0, 106, 0, 152, 0, 174, 0, 118, 0, 264, 0, 0, 0, 270, 0, 184, 0, 0, 0, 142, 0, 312

Offset: 1

Amiram Eldar, Apr 29 2023

nonn

			a(1) = 2 since there are two solutions to iphi(x) = 1: 1 and 2, and 2 is the larger of them.
a(6) = 24 since there are four solutions to iphi(x) = 6: 7, 12, 14 and 24, and 24 is the largest of them.

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

The infinitary version of A057635.

Cf. A091732, A362484, A362486 (positions of 0's), A362667 (record values), A362668 (indices of records).

a[n_] := If[(inv = invIPhi[n]) == {}, 0, Max[inv]]; Array[a, 100] (* using the function invIPhi from A362484 *)

a(A362486(n)) = 0.

A362667 Infinitary sparsely totient numbers: numbers k such that m > k implies iphi(m) > iphi(k), where iphi is the infinitary totient function A091732.

Original entry on oeis.org

2, 6, 8, 10, 24, 30, 42, 54, 56, 66, 120, 168, 216, 264, 270, 312, 330, 384, 408, 456, 480, 510, 552, 840, 1080, 1320, 1560, 1920, 2040, 2280, 2376, 2760, 3000, 3192, 3480, 3720, 3864, 4440, 4920, 5160, 5208, 5640, 7560, 9240, 10920, 11880, 13440, 14280, 15960

Offset: 1

Amiram Eldar, Apr 29 2023

nonn

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

The infinitary version of A036913.
Record values of A362666.
Cf. A091732, A362484, A362668.

s[n_] := If[(inv = invIPhi[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 invIPhi from A362484 *)

A362668 a(n) = A091732(A362667(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 18, 20, 24, 36, 48, 60, 64, 72, 80, 90, 96, 108, 120, 128, 132, 144, 192, 240, 288, 360, 384, 432, 480, 528, 576, 648, 672, 720, 792, 864, 960, 1008, 1080, 1104, 1152, 1440, 1728, 1920, 2160, 2304, 2592, 2880, 3072, 3168, 3456, 3600, 3840

Offset: 1

Amiram Eldar, Apr 29 2023

nonn

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

The infinitary version of A036912.
Indices of records of A362666.
Cf. A091732, A362484, A362667.

s[n_] := If[(inv = invIPhi[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[4000] (* using the function invIPhi from A362484 *)

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 *)

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)).

A362489 a(n) is the least number k such that the equation iphi(x) = k has exactly 2*n solutions, or -1 if no such k exists, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

5, 1, 6, 12, 36, 24, 396, 48, 216, 96, 528, 144, 384, 2784, 432, 240, 1296, 288, 1584, 1800, 480, 1680, 1080, 864, 576, 3240, 2016, 960, 6624, 720, 1152, 7776, 12000, 8448, 5280, 1728, 10752, 2304, 4032, 4800, 6048, 3840, 2160, 5184, 4608, 6336, 1440, 10560, 29568

Offset: 0

Amiram Eldar, Apr 22 2023

nonn

a(n) is the least number k such that A362485(k) = 2*n. Odd values of A362485 are impossible.

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

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

Cf. A091732, A362484, A362485.

Similar sequences: A007374, A063507, A361970, A362186.

solnum[n_] := Length[invIPhi[n]]; seq[len_, kmax_] := Module[{s = Table[-1, {len}], c = 0, k = 1, ind}, While[k < kmax && c < len, ind = solnum[k]/2 + 1; If[ind <= len && s[[ind]] < 0, c++; s[[ind]] = k]; k++]; s]; seq[50, 10^5] (* using the function invIPhi from A362484 *)

A362665 a(n) is the smaller of the two solutions to A091732(x) = A362664(n).

Original entry on oeis.org

1, 3, 4, 5, 11, 16, 33, 23, 29, 43, 69, 64, 47, 53, 76, 87, 59, 71, 79, 83, 141, 101, 103, 159, 107, 177, 131, 137, 213, 149, 163, 249, 167, 173, 236, 179, 235, 191, 197, 303, 309, 265, 321, 253, 223, 227, 229, 316, 239, 332, 251, 256, 393, 263, 269, 411, 283

Offset: 1

Amiram Eldar, Apr 29 2023

nonn

The larger solution is 2*a(n).

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

Cf. A091732, A362484, A362664.

Similar sequences: A131826, A362211, A362212.

invIPhi[#][[1]]& /@ Select[Range[300], Length[invIPhi[#]] == 2 &] (* using the function invIPhi from A362484 *)

cp's OEIS Frontend

A362485 Number of numbers k such that iphi(k) = n, where iphi is the infinitary totient function A091732.

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

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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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A362666 a(n) is the largest m such that iphi(m) = n, where iphi is the infinitary totient function A091732, or a(n) = 0 if no such m exists.

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A362667 Infinitary sparsely totient numbers: numbers k such that m > k implies iphi(m) > iphi(k), where iphi is the infinitary totient function A091732.

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A362668 a(n) = A091732(A362667(n)).

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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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A362488 Record values in A362487.

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A362489 a(n) is the least number k such that the equation iphi(x) = k has exactly 2*n solutions, or -1 if no such k exists, where iphi is the infinitary totient function A091732.

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