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 1-6 of 6 results.

A306984 Infinitary weird numbers: infinitary abundant numbers (A129656) that are not infinitary pseudoperfect numbers (A306983).

Original entry on oeis.org

70, 4030, 5390, 5830, 7192, 7400, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 11830, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17920, 17990, 18410, 18830, 18970, 19390, 19670
Offset: 1

Views

Author

Amiram Eldar, Mar 18 2019

Keywords

Comments

Differs from bi-unitary weird numbers from n >= 32 (a(32) = 17920 is not bi-unitary weird).

Links

Crossrefs

Cf. A006037, A049417, A064114, A126168, A129656, A292986, A321146, A306983.

Programs

  • Mathematica
    idivs[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[ x ] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] ;s = {}; Do[d = Most[idivs[n]]; If[Total[d]

A335935 Infinitary pseudoperfect numbers (A306983) whose number of divisors is not a power of 2.

Original entry on oeis.org

60, 72, 90, 96, 150, 294, 360, 420, 480, 486, 504, 540, 600, 630, 660, 672, 726, 756, 780, 792, 864, 924, 936, 960, 990, 1014, 1020, 1050, 1056, 1092, 1120, 1140, 1152, 1170, 1176, 1188, 1224, 1248, 1344, 1350, 1368, 1380, 1386, 1400, 1428, 1440, 1470, 1500, 1530
Offset: 1

Views

Author

Amiram Eldar, Jun 30 2020

Keywords

Comments

Pseudoperfect numbers (A005835) whose number of divisors is a power of 2 (A036537) are also infinitary pseudoperfect numbers (A306983), since all of their divisors are infinitary.
First differs from A335198 at n = 77.

Examples

			60 is a term since its number of divisors is 12 which is not a power of 2, so not all of its divisors are infinitary, and it is the sum of its infinitary divisors: 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60.

Links

Crossrefs

Subsequence of A005835 and A306983.
Cf. A036537, A077609, A335198.

Programs

  • Mathematica
    idivs[x_] := If[x == 1, 1, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infpspQ[n_] := Module[{d = Most @ idivs[n], x}, Plus @@ d >= n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0]; pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[2, 500], !pow2Q[DivisorSigma[0,#]] && infpspQ[#] &]

A335937 Infinitary pseudoperfect numbers (A306983) that equal to the sum of a subset of their aliquot infinitary divisors in a single way.

Original entry on oeis.org

6, 60, 72, 78, 88, 90, 96, 102, 104, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 426, 438, 474, 486, 498, 534, 582, 606, 618, 642, 654, 678, 726, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1014, 1038, 1074, 1086, 1146, 1158, 1182, 1194
Offset: 1

Views

Author

Amiram Eldar, Jun 30 2020

Keywords

Examples

			72 is a term since its set of infinitary aliquot divisors is {1, 2, 4, 8, 9, 18, 36}, and {1, 8, 9, 18, 36} is its only subset whose sum is equal to 72.

Links

Crossrefs

The infinitary version of A064771.
Subsequence of A306983.
A007357 is a subsequence.
Similar sequences: A295829, A295830, A321145.
Cf. A049417, A077609, A335199.

Programs

  • Mathematica
    idivs[x_] := If[x == 1, 1, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infpspQ[n_] := Module[{d = Most @ idivs[n], x}, Plus @@ d >= n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 1]; Select[Range[2, 1200], infpspQ]

A327945 Nonunitary pseudoperfect numbers: numbers that are equal to the sum of a subset of their nonunitary divisors.

Original entry on oeis.org

24, 36, 48, 72, 80, 96, 108, 112, 120, 144, 160, 168, 180, 192, 200, 216, 224, 240, 252, 264, 288, 300, 312, 320, 324, 336, 352, 360, 384, 392, 396, 400, 408, 416, 432, 448, 456, 468, 480, 504, 528, 540, 552, 560, 576, 588, 600, 612, 624, 640, 648, 672, 684
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2019

Keywords

Comments

The nonunitary version of A005835.

Examples

			36 is in the sequence since its nonunitary divisors are 2, 3, 6, 12, 18 and 36 = 6 + 12 + 18.

Links

Crossrefs

Supersequence of A064591.
Cf. A005835, A064597, A293188, A292985, A306983.

Programs

  • Mathematica
    nudiv[n_] := Module[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; s = {}; Do[d = nudiv[n]; If[Total[d] < n, Continue[]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[s, n]], {n, 1, 700}]; s

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]

A335936 Infinitary weird numbers (A306984) whose number of divisors is not a power of 2.

Original entry on oeis.org

5390, 7400, 11830, 17920, 20230, 25270, 37030, 43750, 58870, 67270, 95830, 117670, 129430, 154630, 168070, 196630, 243670, 260470, 314230, 352870, 373030, 436870, 459270, 482230, 554470, 658630, 714070, 742630, 801430, 831670, 893830, 1024870, 1129030, 1201270
Offset: 1

Views

Author

Amiram Eldar, Jun 30 2020

Keywords

Comments

Weird numbers (A006037) whose number of divisors is a power of 2 (A036537) are also infinitary weird numbers (A306983), since all of their divisors are infinitary.

Links

Crossrefs

Intersection of A162643 and A306984.
Cf. A006037, A036537, A077609, A335935.

Programs

  • Mathematica
    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]; infabQ[n_] := isigma[n] > 2*n; idivs[x_] := If[x == 1, 1, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infwQ[n_] := infabQ[n] && Module[{d = Most @ idivs[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; pow2Q[n_] := n == 2^IntegerExponent[n, 2]; seq = {}; Do[If[!pow2Q[DivisorSigma[0, n]] && infwQ[n], AppendTo[sm n]], {n, 1, 10^5}]; s

Extensions

More terms from Amiram Eldar, Mar 25 2023
Showing 1-6 of 6 results.

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