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A321146 Exponential weird numbers: numbers that are exponential abundant (A129575) but not exponential pseudoperfect (A318100). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The exponential version of A006037.

LINKS

Robert Israel, Table of n, a(n) for n = 1..1000

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

Cf. A006037, A129575, A318100.

Sequence in context: A251325 A186478 A220059 * A107545 A256363 A031568

Adjacent sequences:  A321143 A321144 A321145 * A321147 A321148 A321149

KEYWORD

nonn

AUTHOR

Amiram Eldar, Oct 28 2018

STATUS

approved

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Last modified July 12 06:21 EDT 2020. Contains 335658 sequences. (Running on oeis4.)