A391139 Primitive coreful Zumkeller numbers (second definition): coreful Zumkeller numbers (A339979) that are powerful numbers (A001694).

36, 72, 144, 200, 288, 324, 392, 400, 576, 648, 784, 800, 900, 1152, 1296, 1568, 1600, 1764, 1800, 1936, 2304, 2592, 2700, 2704, 2916, 3136, 3200, 3528, 3600, 3872, 4356, 4500, 4608, 4900, 5000, 5184, 5292, 5400, 5408, 5832, 6084, 6272, 6400, 7056, 7200, 7744, 8100

Offset

1,1

Comments

These are the primitive terms in A339979: Any term in A339979 is of the form k*m where k is a term in this sequence and m is a squarefree number coprime to k.

Therefore, A339979 can be generated from this sequence by multiplying terms with coprime squarefree numbers, and the asymptotic density of A339979 can be evaluated from the terms in this sequence (see the Comments section of A339979).

A number k is a term in A339979 if and only if k divided by its squarefree kernel is a Zumkeller number (A083207). Therefore, this is the sorted sequence of the Zumkeller numbers multiplied by their squarefree kernels (A007947). A number k is a term if and only if there is a Zumkeller number m such that A064549(m) = k.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Mathematica

pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
seq[max_] := Select[pows[max], corZumQ]; seq[10000] (* using the function "corZumQ" from A339979 *)

Crossrefs

Intersection of A001694 and A339979.

Cf. A003557, A005117, A007947, A064549, A083207, A339981 (first definition).

Similar sequences: A391138, A391140, A391141.

Sequence in context: A119843 A339979 A322657 * A364930 A339980 A347892

Adjacent sequences: A391136 A391137 A391138 * A391140 A391141 A391142

Keyword

nonn

Author

Amiram Eldar, Dec 01 2025

Status

approved