OFFSET
1,1
COMMENTS
For squarefree numbers k, eusigma(k) = eisigma(k) = k, where eusigma is the sum of exponential unitary divisors function (A322857), and eisigma is the sum of exponential infinitary divisors function (A361175). Thus, if m is a term (eisigma(m) > 2*m >= eusigma(m)) and k is a squarefree number coprime to m, then eusigma(k*m) = eusigma(k) * eusigma(m) = k * eusigma(m) <= 2*k*m, and eisigma(k*m) = eisigma(k) * eisigma(m) = k * eisigma(m) > 2*k*m, so k*m is an exponential infinitary abundant number that is not exponential unitary abundant (A383695). Therefore, the sequence A383695 can be generated from this sequence by multiplying with coprime squarefree numbers.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
MATHEMATICA
idivs[1] = {1}; idivs[n_] := Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
fi[p_, e_] := Total[p^idivs[e]]; fu[p_, e_] := DivisorSum[e, p^# &, CoprimeQ[#, e/#] &];
q[n_] := Module[{fct = FactorInteger[n]}, Times @@ fu @@@ fct <= 2*n < Times @@ fi @@@ fct];
pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
seqA383696[max_] := Select[pows[max], q]; seqA383696[10^10]
PROG
(PARI) isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); } \\ Michel Marcus at A077609
fi(p, e) = sumdiv(e, d, if(isidiv(d, factor(e)), p^d, 0));
fu(p, e) = sumdiv(e, d, if(gcd(d, e/d)==1, p^d));
isprim(k) = {my(f = factor(k)); prod(i = 1, #f~, fu(f[i, 1], f[i, 2])) <= 2*k && prod(i = 1, #f~, fi(f[i, 1], f[i, 2])) > 2*k; }
listpows(lim) = my(v = List(), t); for(m = 1, sqrtnint(lim\1, 3), t=m^3; for(n = 1, sqrtint(lim\t), listput(v, t*n^2))); Set(v) \\ Charles R Greathouse IV at A001694
listA383696(lim) = select(x -> isprim(x), listpows(lim));
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, May 06 2025
STATUS
approved