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A383699 Primitive exponential 3-abundant numbers: the powerful terms of A328135.

%I A383699 #10 May 07 2025 10:50:52
%S A383699 901800900,1542132900,1926332100,2153888100,2690496900,2822796900,
%T A383699 3942584100,4487660100,4600908900,5127992100,6267888900,6742052100,
%U A383699 7162236900,7305120900,8421732900,8969984100,9866448900,10203020100,10718460900,11723411700,11787444900,12528324900
%N A383699 Primitive exponential 3-abundant numbers: the powerful terms of A328135.
%C A383699 Subsequence of A328135 and first differ from it at n = 25: A328135(25) = 15330615300 is not a term of this sequence.
%C A383699 For squarefree numbers k, esigma(k) = k, where esigma is the sum of exponential divisors function (A051377). Thus, if m is a term (esigma(m) >= 3*m) and k is a squarefree number coprime to m, then esigma(k*m) = esigma(k) * esigma(m) = k * esigma(m) >= 3*k*m, so k*m is an exponential 3-abundant number. Therefore, the sequence of exponential 3-abundant numbers (A328135) can be generated from this sequence by multiplying with coprime squarefree numbers.
%H A383699 Amiram Eldar, <a href="/A383699/b383699.txt">Table of n, a(n) for n = 1..10136</a> (terms below 10^16)
%e A383699 901800900 is a term since esigma(901800900) = 2905943040 > 3 * 901800900 = 2705402700, and 901800900 = 2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 13^2 is a powerful number.
%t A383699 pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
%t A383699 f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n];
%t A383699 seq[max_] := Select[pows[max], esigma[#] >= 3 # &]; seq[10^10]
%Y A383699 Intersection of A001694 and A328135.
%Y A383699 Subsequence of A328136.
%Y A383699 Cf. A051377, A307112.
%K A383699 nonn
%O A383699 1,1
%A A383699 _Amiram Eldar_, May 06 2025