A359417 -id:A359417 - cp's OEIS frontend

A359418 Unitary phi-practical (A286906) whose unitary divisors have distinct values of the unitary totient function uphi (A047994).

1, 3, 15, 105, 165, 195, 255, 1155, 1785, 1995, 2145, 2415, 2805, 3045, 3135, 3255, 3315, 3705, 3795, 3885, 4305, 4485, 4515, 4785, 4845, 4935, 5115, 5565, 5655, 5865, 6045, 6105, 6195, 6405, 7035, 7095, 7215, 7395, 7455, 7665, 7755, 7905, 7995, 8295, 8385, 8715

Offset: 1

Amiram Eldar, Dec 31 2022

nonn

A unitary phi-practical number k is a number k such that each number in the range 1..k is a subsum of a the multiset {uphi(d) : d | k, gcd(d, k/d) = 1}. This sequence is restricted to cases in which all the values in this multiset are distinct.

Are all the terms above 3 divisible by 5?

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

Intersection of A286906 and A348004.

Cf. A047994, A359417.

uphi[n_] := If[n == 1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]];
uDivisors[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; uPhiPracticalQ[n_] := If[n < 1, False, If[n == 1, True, (lst = Sort @ Map[uphi, uDivisors[n]]; ok = True; Do[If[lst[[m]] > Sum[lst[[l]], {l, 1, m - 1}] + 1, (ok = False; Break[])], {m, 1, Length[lst]}]; ok)]];
Select[Range[9000], UnsameQ @@ uphi /@ Divisors[#] && uPhiPracticalQ[#] &]

A361925 Infinitary phi-practical (A361922) whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

Original entry on oeis.org

1, 3, 12, 15, 60, 105, 108, 132, 156, 165, 195, 240, 255, 660, 960, 1020, 1140, 1155, 1188, 1380, 1680, 1716, 1728, 1740, 1785, 1860, 1995, 2052, 2145, 2220, 2244, 2415, 2460, 2484, 2496, 2508, 2580, 2640, 2652, 2805, 2820, 2940, 3036, 3045, 3120, 3132, 3135, 3180

Offset: 1

Amiram Eldar, Mar 30 2023

nonn

An infinitary phi-practical number k is a number k such that each number in the range 1..k is a subsum of the multiset {iphi(d) | d infinitary divisor of k}. This sequence is restricted to cases in which all the values in this multiset are distinct.

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

Intersection of A361922 and A361924.
Cf. A091732.
Similar sequences: A359417, A359418.

f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
iPhiPracticalQ[n_] := Module[{s = Sort@Map[iphi, idivs[n]], ans = True}, Do[If[s[[j]] > Sum[s[[i]], {i, 1, j - 1}] + 1, ans = False; Break[]], {j, 1, Length[s]}]; ans];
Select[Range[3200], UnsameQ @@ iphi /@ idivs[#] && iPhiPracticalQ[#] &]

cp's OEIS Frontend

A359418 Unitary phi-practical (A286906) whose unitary divisors have distinct values of the unitary totient function uphi (A047994).

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