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.

A286652 Unitary practical numbers: numbers n such that every 1 <= k <= usigma(n) is a sum of distinct unitary divisors of n.

Original entry on oeis.org

1, 2, 6, 30, 42, 66, 78, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1050, 1110, 1122, 1218, 1230, 1254, 1290, 1302, 1326, 1410, 1470, 1482, 1518, 1554, 1590, 1650, 1722, 1770, 1794, 1806, 1830, 1914, 1950, 1974, 2010, 2046, 2130
Offset: 1

Views

Author

Amiram Eldar, May 27 2017

Keywords

Comments

The unitary version of A005153. The squarefree terms of both sequences are the same, A265501. The nonsquarefree terms of this sequence are in A287173.

Links

Crossrefs

Cf. A005153, A034448, A265501, A287173.

Programs

  • Mathematica
    usigma[n_] :=  Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[#, n/#] == 1 &]]; uPracticalQ[n_] :=  Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, f = FactorInteger[n]; {p, e} = Transpose[f]; r = Sort[p^e]; Do[If[r[[i]] > 1 + usigma[prod], ok = False; Break[]]; prod = prod*r[[i]], {i, Length[p]}]; ok]]]; Select[ Range[100000], uPracticalQ]

A334899 Bi-unitary practical numbers (A334898) that are not exponentially odd numbers (A268335).

Original entry on oeis.org

48, 72, 192, 240, 288, 320, 336, 360, 432, 448, 504, 528, 600, 624, 648, 768, 792, 800, 810, 816, 912, 936, 960, 1050, 1104, 1134, 1152, 1176, 1200, 1224, 1280, 1296, 1344, 1350, 1368, 1392, 1400, 1440, 1470, 1488, 1568, 1650, 1656, 1680, 1728, 1776, 1782, 1792
Offset: 1

Views

Author

Amiram Eldar, May 16 2020

Keywords

Comments

Practical numbers (A005153) that are exponentially odd (A268335) are also bi-unitary practical numbers (A334898), since all of their divisors are bi-unitary.
Of the first 2500 bi-unitary practical numbers, only 847 are in this sequence.

Links

Crossrefs

Cf. A005153, A268335, A287173, A334898, A334902.

Programs

  • Mathematica
    biunitaryDivisorQ[div_, n_] := If[Mod[#2, #1] == 0, Last @ Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]] &, {#1, #2/#1}]] == 1, False] & @@ {div, n}; bdivs[n_] := Module[{d = Divisors[n]}, Select[d, biunitaryDivisorQ[#, n] &]]; bPracQ[n_] := Module[{d = bdivs[n], sd, x}, sd = Plus @@ d; Min @ CoefficientList[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, sd}], x] >  0]; expOddQ[n_] := AllTrue[Last /@ FactorInteger[n], OddQ]; Select[Range[1000], !expOddQ[#] && bPracQ[#] &]

A334902 Infinitary practical numbers (A334901) whose number of divisors is not a power of 2.

Original entry on oeis.org

72, 360, 480, 504, 600, 672, 792, 864, 936, 1050, 1056, 1152, 1176, 1224, 1248, 1350, 1368, 1400, 1470, 1650, 1656, 1800, 1950, 1960, 2088, 2200, 2232, 2520, 2600, 2646, 2664, 2952, 3096, 3200, 3234, 3240, 3360, 3384, 3402, 3528, 3816, 3822, 3960, 4200, 4248, 4312
Offset: 1

Views

Author

Amiram Eldar, May 16 2020

Keywords

Comments

Practical numbers (A005153) whose number of divisors is a power of 2 (A036537) are also infinitary practical numbers (A334901), since all of their divisors are infinitary.
Up to 10^6 there are 34768 infinitary practical numbers; of them only 8858 are in this sequence.

Links

Crossrefs

Cf. A005153, A036537, A287173, A334899, A334901.

Programs

  • Mathematica
    bin[n_] := 2^(-1 + Position[Reverse @ IntegerDigits[n, 2], ?(# == 1 &)] // Flatten); f[p, e_] := p^bin[e]; icomp[n_] := Flatten[f @@@ FactorInteger[n]]; fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; infPracQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, r = Sort[icomp[n]]; Do[If[r[[i]] > 1 + isigma[prod], ok = False; Break[]]; prod = prod*r[[i]], {i, Length[r]}]; ok]]]; pow2Q[n_] := n/2^IntegerExponent[n, 2] == 1; Select[Range[4400], ! pow2Q[DivisorSigma[0, #]] && infPracQ[#] &]
Showing 1-3 of 3 results.

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