A321145 Exponential pseudoperfect numbers (A318100) equal to the sum of a subset of their proper exponential divisors in a single way.

36, 180, 252, 396, 468, 612, 684, 828, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4500, 4572, 4716, 4788, 4932

Offset: 1

Author

Amiram Eldar, Oct 28 2018

Keywords

• nonn

Comments

The exponential version of A064771.

Examples

			4500 is in the sequence since its proper exponential divisors are 30, 60, 90, 180, 750, 1500, 2250 and {750, 1500, 2250} is the only subset that sums to 4500.
		

Links

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

Crossrefs

Cf. A064771, A295829, A295830, A318100.

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 == 1, AppendTo[a, n]]]; a