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

%I A318100 #33 Feb 16 2025 08:33:56
%S A318100 36,180,252,396,468,612,684,828,900,1044,1116,1260,1332,1476,1548,
%T A318100 1692,1764,1800,1908,1980,2124,2196,2340,2412,2556,2628,2700,2772,
%U A318100 2844,2988,3060,3204,3276,3420,3492,3600,3636,3708,3852,3924,4068,4140,4284,4356,4500
%N A318100 Exponential pseudoperfect numbers: numbers n equal to the sum of a subset of their proper exponential divisors.
%H A318100 Charles R Greathouse IV, <a href="/A318100/b318100.txt">Table of n, a(n) for n = 1..10000</a>
%H A318100 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>
%H A318100 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-PerfectNumber.html">e-Perfect Number</a>
%e A318100 900 is in the sequence since its proper exponential divisors are 30, 60, 90, 150, 180, 300, 450 and 900 = 150 + 300 + 450.
%t A318100 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
%o A318100 (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)
%o A318100 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
%Y A318100 The exponential version of A005835. A054979 is a subsequence.
%Y A318100 Cf. A051377, A126164, A129575, A292985, A293188.
%K A318100 nonn
%O A318100 1,1
%A A318100 _Amiram Eldar_, Oct 28 2018