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 1-4 of 4 results.

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[#] &]

A334903 Numbers k such that k and k+2 are both infinitary practical numbers (A334901).

Original entry on oeis.org

6, 40, 54, 918, 1240, 1288, 1408, 1480, 1672, 1720, 1768, 1974, 2440, 2728, 2838, 2968, 3198, 3318, 4134, 4264, 4422, 4480, 4758, 5248, 6102, 6270, 6424, 7590, 7830, 10624, 11128, 13110, 13182, 14248, 15496, 15928, 16254, 16768, 18088, 19864, 21112, 21318, 21630
Offset: 1

Views

Author

Amiram Eldar, May 16 2020

Keywords

Examples

			6 is a term since 6 and 6 + 2 = 8 are both infinitary practical numbers.

Links

Crossrefs

Cf. A287681, A330871, A334882, A334900, 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]]]; seq = {}; q1 = infPracQ[2]; Do[q2 = infPracQ[n]; If[q1 && q2, AppendTo[seq, n - 2]]; q1 = q2, {n, 4, 10^4, 2}]; seq

A334898 Bi-unitary practical numbers: numbers m such that every number 1 <= k <= bsigma(m) is a sum of distinct bi-unitary divisors of m, where bsigma is A188999.

Original entry on oeis.org

1, 2, 6, 8, 24, 30, 32, 40, 42, 48, 54, 56, 66, 72, 78, 88, 96, 104, 120, 128, 160, 168, 192, 210, 216, 224, 240, 264, 270, 280, 288, 312, 320, 330, 336, 352, 360, 378, 384, 390, 408, 416, 432, 440, 448, 456, 462, 480, 486, 504, 510, 512, 520, 528, 544, 546, 552
Offset: 1

Views

Author

Amiram Eldar, May 16 2020

Keywords

Comments

Includes 1 and all the odd powers of 2 (A004171). The other terms are a subset of bi-unitary abundant numbers (A292982) and bi-unitary pseudoperfect numbers (A292985).

Links

Crossrefs

The bi-unitary version of A005153.
Cf. A004171, A188999, A286652, A292982, A292985, A334901.

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]; Select[Range[1000], bPracQ]

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]
Showing 1-4 of 4 results.

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