A383698 Primitive exponential squarefree exponential abundant numbers: the powerful terms of A383697.

900, 1764, 4356, 4500, 4900, 6084, 10404, 12348, 12996, 19044, 30276, 34596, 44100, 47916, 49284, 60516, 66564, 79092, 79524, 88200, 101124, 108900, 112500, 125316, 132300, 133956, 152100, 161604, 176868, 181476, 191844, 213444, 217800, 220500, 224676, 246924

Offset: 1

Author

Amiram Eldar, May 06 2025

Keywords

• nonn

Comments

Subsequence of A383694 and first differs from it at n = 11.

The least odd term is a(1345) = 225450225, and the least term that is coprime to 6 is 1117347505588495206025.

For squarefree numbers k, essigma(k) = k, where essigma is the sum of exponential squarefree exponential divisors function (A361174). Thus, if m is a term (essigma(m) > 2*m) and k is a squarefree number coprime to m, then essigma(k*m) = essigma(k) * essigma(m) = k * essigma(m) > 2*k*m, so k*m is an exponential squarefree exponential abundant number. Therefore, the sequence of exponential squarefree exponential abundant numbers (A383697) 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 A383697.

Subsequence of A383694.

Cf. A005117, A361174.

Programs

Mathematica

fun[p_, e_] := DivisorSum[e, p^# &, SquareFreeQ[#] &]; q[n_] := Min[(f = FactorInteger[n])[[;; , 2]]] > 1 && Times @@ fun @@@ f > 2*n; Select[Range[250000], q]

PARI

fun(p, e) = sumdiv(e, d, if(issquarefree(d), p^d, 0));
isok(k) = {my(f = factor(k)); ispowerful(f) && prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}