A383699 Primitive exponential 3-nondeficient 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

1,1

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-nondeficient number. Therefore, the sequence of exponential 3-nondeficient numbers (A328135) can be generated from this sequence by multiplying with coprime squarefree numbers.

Links

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

Example

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.

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]

Crossrefs

Intersection of A001694 and A328135.

Subsequence of A328136.

Cf. A051377, A307112.

Sequence in context: A178557 A157798 A328135 * A189229 A051470 A076135

Adjacent sequences: A383696 A383697 A383698 * A383700 A383701 A383702

Keyword

nonn

Author

Amiram Eldar, May 06 2025

Extensions

Name corrected by Amiram Eldar, Sep 24 2025

Status

approved