OFFSET
1,1
COMMENTS
From Amiram Eldar, Dec 01 2025: (Start)
If k is a term and m is a squarefree number coprime to k, then k*m is also a term. The primitive terms in this sequence (A391143) are the powerful (A001694) terms. All the terms are of the form k*m where k is primitive and m is a squarefree number coprime to k.
The asymptotic density of this sequence is Sum_{n>=1} f(A391143(n)) = 0.0096773..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)).
The least odd term is a(2181735) = 225450225 = (3 * 5 * 7 * 11 * 13)^2. (End)
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.
EXAMPLE
900 is in the sequence since its proper exponential divisors are 30, 60, 90, 150, 180, 300, 450 and 900 = 150 + 300 + 450.
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] < 2*n;
ePspQ[n_] := !eDeficientQ[n] && Module[{d = Most[eDivs[n]], x}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 1, Length[d]}], {x, 0, n}], n] > 0];
Select[Range[4500], ePspQ] (* updated on Dec 01 2025 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 28 2018
STATUS
approved
