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

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

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)

Eric Weisstein's World of Mathematics, e-Divisor.

Eric Weisstein's World of Mathematics, e-Perfect Number.

Example

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.

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

Crossrefs

The exponential version of A006037.

Cf. A129575, A318100.

Sequence in context: A186478 A220059 A339939 * A107545 A256363 A031568

Adjacent sequences: A321143 A321144 A321145 * A321147 A321148 A321149

Keyword

nonn

Author

Amiram Eldar, Oct 28 2018

Status

approved