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

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

Offset: 1

Author

Amiram Eldar, Oct 28 2018

Keywords

• nonn

Examples

			900 is in the sequence since its proper exponential divisors are 30, 60, 90, 150, 180, 300, 450 and 900 = 150 + 300 + 450.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, e-Divisor
• Eric Weisstein's World of Mathematics, e-Perfect Number

Crossrefs

The exponential version of A005835. A054979 is a subsequence.

Cf. A051377, A126164, A129575, A292985, A293188.

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 > 0, AppendTo[a, n]]]; a

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)
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