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A321146
Exponential weird numbers: numbers that are exponential abundant (A129575) but not exponential pseudoperfect (A318100).
7
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.
Sequence in context: A186478 A220059 A339939 * A374500 A107545 A256363
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 28 2018
STATUS
approved