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

901800900, 1542132900, 1926332100, 2153888100, 2690496900, 2822796900, 3942584100, 4487660100, 4600908900, 5127992100, 6267888900, 6742052100, 7162236900, 7305120900, 8421732900, 8969984100, 9866448900, 10203020100, 10718460900, 11723411700, 11787444900, 12528324900

Offset: 1

Author

Amiram Eldar, May 06 2025

Keywords

• nonn

Comments

Subsequence of A328135 and first differ from it at n = 25: A328135(25) = 15330615300 is not a term of this sequence.

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.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10136 (terms below 10^16)

Crossrefs

Intersection of A001694 and A328135.

Subsequence of A328136.

Cf. A051377, A307112.

Programs

Mathematica

pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n];
seq[max_] := Select[pows[max], esigma[#] >= 3 # &]; seq[10^10]