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 1-7 of 7 results.

A292986 Bi-unitary weird numbers: bi-unitary abundant numbers (A292982) that are not bi-unitary pseudoperfect (A292985).

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, 17990, 18410, 18830, 18970, 19390, 19670, 19810
Offset: 1

Views

Author

Amiram Eldar, Sep 27 2017

Keywords

Comments

Analogous to weird numbers (A006037) with bi-unitary sigma (A188999) instead of sigma (A000203).

Links

Crossrefs

Cf. A006037, A064114, A188999, A292982, A292985.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bAbundantQ[n_] := bsigma[n] > 2 n; a = {}; n = 0; While[Length[a] < 5, n++; If[!bAbundantQ[n], Continue[]]; d = Most[bdiv[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c <= 0, AppendTo[a, n]]]; a (* after T. D. Noe at A005835 and Michael De Vlieger at A188999 *)

A335938 Bi-unitary pseudoperfect numbers (A292985) that are not exponentially odd numbers (A268335).

Original entry on oeis.org

48, 60, 72, 80, 90, 150, 162, 192, 240, 288, 294, 320, 336, 360, 420, 432, 448, 504, 528, 540, 560, 576, 600, 624, 630, 648, 660, 704, 720, 726, 756, 768, 780, 792, 800, 810, 816, 832, 880, 912, 924, 936, 960, 990, 1008, 1014, 1020, 1040, 1050, 1092, 1104, 1134
Offset: 1

Views

Author

Amiram Eldar, Jun 30 2020

Keywords

Comments

Pseudoperfect numbers (A005835) that are exponentially odd (A268335) are also bi-unitary pseudoperfect numbers (A292985), since all of their divisors are bi-unitary.
First differs from A335216 at n = 28.

Examples

			48 is a term since it is not exponentially odd number (48 = 2^4 * 3 and 4 is even), so not all of its divisors are bi-unitary, and it is the sum of a subset of its bi-unitary divisors: 8 + 16 + 24 = 48.

Links

Crossrefs

Subsequence of A005835 and A292985.
Cf. A222266, A268335, A295830, A335216.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; bPspQ[n_] := Module[{d = Most @ bdiv[n], x}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0]; expOddQ[n_] := AllTrue[Last /@ FactorInteger[n], OddQ]; Select[Range[1000], ! expOddQ[#] && bPspQ[#] &]

A306983 Infinitary pseudoperfect numbers: numbers n equal to the sum of a subset of their proper infinitary divisors.

Original entry on oeis.org

6, 24, 30, 40, 42, 54, 56, 60, 66, 72, 78, 88, 90, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498, 504
Offset: 1

Views

Author

Amiram Eldar, Mar 18 2019

Keywords

Comments

Subsequence of A005835.

Links

Crossrefs

Cf. A005835, A007357, A049417, A126168, A129656, A293188, A292985, A318100, A306984.

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]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[s, n]], {n, 2, 1000}]; s

A318100 Exponential pseudoperfect numbers: numbers n equal to the sum of a subset of their proper exponential divisors.

Original entry on oeis.org

36, 180, 252, 396, 468, 612, 684, 828, 900, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1764, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3600, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4356, 4500
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2018

Keywords

Examples

			900 is in the sequence since its proper exponential divisors are 30, 60, 90, 150, 180, 300, 450 and 900 = 150 + 300 + 450.

Links

Crossrefs

The exponential version of A005835. A054979 is a subsequence.
Cf. A051377, A126164, A129575, A292985, A293188.

Programs

  • Mathematica
    dQ[n_,m_] := (n>0&&m>0 &&Divisible[n,m]); expDivQ[n_,d_] := Module[ {ft=FactorInteger[n]}, And@@MapThread[dQ, {ft[[;;,2]], IntegerExponent[ d,ft[[;;,1]]]} ]]; eDivs[n_] := Module[ {d=Rest[Divisors[n]]}, Select[ d,expDivQ[n,#]&] ]; esigma[1]=1; esigma[n_] := Total@eDivs[n]; eDeficientQ[n_] := esigma[n] < 2n; a = {}; n = 0; While[Length[a] < 30, n++; If[eDeficientQ[n], Continue[]]; d = Most[eDivs[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[a, n]]]; a
  • PARI
    ediv(n,f=factor(n))=my(v=List(),D=apply(divisors,f[,2]~),t=#f~); forvec(u=vector(t,i,[1,#D[i]]), listput(v,prod(j=1,t,f[j,1]^D[j][u[j]]))); Set(v)
is(n)=my(e=ediv(n)); e=e[1..#e-1]; forsubset(#e, v, if(vecsum(vecextract(e,v))==n, return(1))); 0 \\ Charles R Greathouse IV, Oct 29 2018

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

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]

A295830 Bi-unitary pseudoperfect numbers that equal to the sum of a subset of their aliquot bi-unitary divisors in a single way.

Original entry on oeis.org

6, 60, 72, 78, 80, 88, 90, 102, 104, 114, 138, 150, 162, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 704, 726, 762, 786, 822, 832, 834, 894, 906, 942, 978, 1002, 1014, 1038, 1074, 1086, 1146
Offset: 1

Views

Author

Amiram Eldar, Nov 28 2017

Keywords

Comments

The bi-unitary version of A064771.

Examples

			72 is in the sequence since its aliquot bi-unitary divisors are 1, 2, 4, 8, 9, 18, 36 and {1, 8, 9, 18, 36} is the only subset whose sum is 72.

Crossrefs

Cf. A064771, A292985.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; a={}; n=0;While[Length[a]<100,n++;d=Most[bdiv[n]];c = SeriesCoefficient[ Series[ Product[1+x^d[[i]],{i,Length[d]}],{x,0,n}],n ];If[c==1; AppendTo[a,n]]];a
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.