cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Dec 31 2022

Keywords

Comments

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?

Links

Crossrefs

Intersection of A286906 and A348004.
Cf. A047994, A359417.

Programs

  • Mathematica
    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[#] &]

A361922 Infinitary phi-practical numbers: numbers m such that each k <= m is a subsum of a the multiset {iphi(d) : d infinitary divisor of m}, where iphi is an infinitary analog of Euler's phi function (A091732).

Original entry on oeis.org

1, 2, 3, 6, 8, 12, 15, 24, 30, 40, 42, 56, 60, 72, 84, 105, 108, 120, 132, 135, 156, 165, 168, 195, 210, 216, 240, 255, 264, 270, 280, 312, 330, 360, 378, 384, 390, 408, 420, 440, 456, 462, 480, 504, 510, 520, 540, 546, 552, 570, 600, 616, 640, 660, 672, 680, 690
Offset: 1

Views

Author

Amiram Eldar, Mar 30 2023

Keywords

Links

Crossrefs

Cf. A077609, A091732.
Similar sequences: A260653, A286906, A334901.

Programs

  • Mathematica
    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[700], iPhiPracticalQ]

A359419 Nonsquarefree numbers that are both phi-practical and unitary phi-practical.

Original entry on oeis.org

12, 60, 84, 120, 132, 156, 240, 420, 660, 780, 840, 924, 1020, 1050, 1092, 1140, 1320, 1380, 1428, 1560, 1596, 1680, 1716, 1740, 1860, 1932, 2040, 2100, 2220, 2244, 2280, 2436, 2460, 2508, 2580, 2604, 2640, 2652, 2760, 2820, 2940, 2964, 3036, 3108, 3120, 3180
Offset: 1

Views

Author

Amiram Eldar, Dec 31 2022

Keywords

Comments

The squarefree numbers (A005117) are excluded from this sequence since every squarefree phi-practical number is also a unitary phi-practical number.
The least odd term in this sequence is a(104) = 8085.

Links

Crossrefs

Intersection of A013929, A260653 and A286906.
Cf. A005117.

Programs

  • Mathematica
    phiPracticalQ[n_] := If[n == 1, True, (lst = Sort @ EulerPhi @ Divisors[n]; ok = True;  Do[If[lst[[m]] > Sum[lst[[l]], {l, 1, m - 1}] + 1, (ok = False; Break[])], {m, 1, Length[lst]}]; ok)];
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)]];  (* Frank M Jackson's code at A260653 *)
Select[Range[3200], ! SquareFreeQ[#] && phiPracticalQ[#] && uPhiPracticalQ[#] &]
Showing 1-3 of 3 results.

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