A321206 Exponential pseudoperfect numbers (A318100) that are not e-perfect (A054979).

900, 1764, 3600, 4356, 4500, 6084, 6300, 7056, 8100, 8820, 9900, 10404, 11700, 12348, 12996, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 25200, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 35280, 36900, 38700, 39600, 40572

Offset: 1

Author

Amiram Eldar, Oct 30 2018

Keywords

• nonn

Comments

It seems that most of the exponential pseudoperfect numbers are e-perfect. Up to 10^6 there are 9674 exponential pseudoperfect numbers, of them only 984 are not e-perfect.

Links

• Amiram Eldar, 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

Crossrefs

Cf. A054979, 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]; eAbundantQ[n_] := esigma[n] > 2n; 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 > 0, AppendTo[a, n]]]; a