A383696 Primitive exponential infinitary abundant numbers that are not primitive exponential unitary abundant: the powerful terms of A383695.

476985600, 815673600, 1018886400, 1177862400, 1493049600, 2014214400, 2373638400, 2712326400, 3756614400, 3863865600, 4744454400, 5218617600, 6234681600, 7928121600, 9824774400, 10502150400, 12669753600, 14227718400, 15040569600, 17614598400, 19443513600, 22356230400

Offset: 1

Author

Amiram Eldar, May 06 2025

Keywords

• nonn

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

Crossrefs

Intersection of A001694 and A383695.

Cf. A005117, A322857, A361175.

Programs

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]

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));