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.

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[#] &]

A334900 Numbers k such that k and k+2 are both bi-unitary practical numbers (A334898).

Original entry on oeis.org

6, 30, 40, 54, 510, 544, 798, 918, 928, 1120, 1240, 1288, 1408, 1480, 1566, 1672, 1720, 1768, 1792, 1888, 1950, 1974, 2046, 2430, 2440, 2560, 2728, 2814, 2838, 2968, 3198, 3318, 4134, 4158, 4264, 4422, 4480, 4758, 5248, 6102, 6270, 6424, 6942, 7590, 7830, 9280
Offset: 1

Views

Author

Amiram Eldar, May 16 2020

Keywords

Examples

			6 is a term since 6 and 6 + 2 = 8 are both bi-unitary practical numbers.

Links

Crossrefs

Cf. A287681, A330871, A334882, A334898, A334903.

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]; seq = {}; q1 = bPracQ[2]; Do[q2 = bPracQ[n]; If[q1 && q2, AppendTo[seq, n - 2]]; q1 = q2, {n, 4, 1000, 2}]; seq

A334901 Infinitary practical numbers: numbers m such that every number 1 <= k <= isigma(m) is a sum of distinct infinitary divisors of m, where isigma is A049417.

Original entry on oeis.org

1, 2, 6, 8, 24, 30, 40, 42, 54, 56, 66, 72, 78, 88, 104, 120, 128, 168, 210, 216, 264, 270, 280, 312, 330, 360, 378, 384, 390, 408, 440, 456, 462, 480, 504, 510, 520, 546, 552, 570, 594, 600, 616, 640, 672, 680, 690, 696, 702, 714, 728, 744, 750, 760, 792, 798
Offset: 1

Views

Author

Amiram Eldar, May 16 2020

Keywords

Comments

Includes the powers of 2 of the form 2^(2^k - 1) for k = 0, 1, ... (A058891). The other terms are a subset of infinitary abundant numbers (A129656) and infinitary pseudoperfect numbers (A306983).

Links

Crossrefs

The infinitary version of A005153.
Cf. A049417, A058891, A129656, A286652, A306983, A334898.

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]]]; Select[Range[1000], infPracQ]
Showing 1-3 of 3 results.

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