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 1-5 of 5 results.

A321146 Exponential weird numbers: numbers that are exponential abundant (A129575) but not exponential pseudoperfect (A318100).

Original entry on oeis.org

4900, 14700, 53900, 63700, 83300, 93100, 112700, 142100, 151900, 161700, 181300, 191100, 200900, 210700, 230300, 249900, 259700, 279300, 289100, 298900, 328300, 338100, 347900, 357700, 387100, 406700, 426300, 436100, 455700, 475300, 494900, 504700, 524300
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2018

Keywords

Examples

			4900 is in the sequence since its proper exponential divisors, {70, 140, 350, 490, 700, 980, 2450} sum to 5180 > 4900, yet no subset of its divisors sums to 4900.

Links

Crossrefs

The exponential version of A006037.
Cf. A129575, A318100.

Programs

  • Maple
    filter:= proc(n)
  local L,m,P,i,j,T,S,t,v;
  L:= ifactors(n)[2];
  m:= nops(L);
  P:= map(t -> numtheory:-divisors(t[2]),L);
  if mul(add(L[i][1]^j, j=P[i]),i=1..m) <= 2*n then return false fi;
  T:= combinat:-cartprod(P);
  S:= {0}:
  while not T[finished] do
    t:= T:-nextvalue();
    v:= mul(L[i][1]^t[i],i=1..m);
    if v = n then next fi;
    if member(n-v,S) then return false fi;
    S:= S union select(`<=`,map(`+`,S,v),n);
  od;
  true
end proc:
select(filter, [$1..10^6]); # Robert Israel, Feb 19 2019
  • 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]; eAbundantQ[n_] := esigma[n] > 2 n; a = {}; n = 0; While[Length[a] < 30, n++; If[!eAbundantQ[n], Continue[]]; d = Most[eDivs[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c < 1, AppendTo[a, n]]]; a

A321145 Exponential pseudoperfect numbers (A318100) equal to the sum of a subset of their proper exponential divisors in a single way.

Original entry on oeis.org

36, 180, 252, 396, 468, 612, 684, 828, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4500, 4572, 4716, 4788, 4932
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2018

Keywords

Comments

The exponential version of A064771.

Examples

			4500 is in the sequence since its proper exponential divisors are 30, 60, 90, 180, 750, 1500, 2250 and {750, 1500, 2250} is the only subset that sums to 4500.

Links

Crossrefs

Cf. A064771, A295829, A295830, A318100.

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 == 1, AppendTo[a, n]]]; a

A321206 Exponential pseudoperfect numbers (A318100) that are not e-perfect (A054979).

Original entry on oeis.org

900, 1764, 3600, 4356, 4500, 6084, 6300, 7056, 8100, 8820, 9900, 10404, 11700, 12348, 12996, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 25200, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 35280, 36900, 38700, 39600, 40572
Offset: 1

Views

Author

Amiram Eldar, Oct 30 2018

Keywords

Comments

It seems that most of the exponential pseudoperfect numbers are e-perfect. Up to 10^6 there are 9674 exponential pseudoperfect numbers, of them only 984 are not e-perfect.

Links

Crossrefs

Cf. A054979, A318100.

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]; eAbundantQ[n_] := esigma[n] > 2n; a = {}; n = 0; While[Length[a] < 30, n++; If[!eAbundantQ[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

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

A335218 Exponential Zumkeller numbers: numbers whose exponential divisors can be partitioned into two disjoint subsets of equal sum.

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, 4572, 4716, 4788, 4900
Offset: 1

Views

Author

Amiram Eldar, May 27 2020

Keywords

Comments

First differs from A318100 at n = 49: 4900 is a term that is not an exponential pseudoperfect number.

Examples

			36 is a term since its exponential divisors, {6, 12, 18, 36}, can be partitioned into 2 disjoint sets whose sum is equal: 6 + 12 + 18 = 36.

Links

Crossrefs

The exponential version of A083207.
Subsequence of A129575.
A054979 is a subsequence.
Cf. A318100, A322791, A323343.

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, #] &]]; ezQ[n_] := Module[{d = eDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; Select[Range[10^4], ezQ]
Showing 1-5 of 5 results.

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