A362486 -id:A362486 - 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.

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.

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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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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Mathematica

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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