A346014 -id:A346014 - cp's OEIS frontend

A346015 Least integers of their prime signature (A025487) whose average number of distinct prime factors of their divisors is an integer.

1, 6, 210, 900, 1080, 1440, 30030, 69300, 83160, 110880, 5292000, 9261000, 9525600, 9699690, 15315300, 18378360, 24504480, 67737600, 170100000, 181440000, 244944000, 416215800, 435456000, 554954400, 756756000, 901800900, 1324323000, 1362160800, 6469693230, 6692786100

Offset: 1

Amiram Eldar, Jul 01 2021

nonn

Since f(k) = A346009(k)/A346010(k), the average number of distinct prime factors of the divisors of k, depends only on the prime signature of k, the sequence A346014 consists of all the numbers with the same prime signature of terms of this sequence.

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

Intersection of A025487 and A346014.

Cf. A346009, A346010.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; f[p_, e_] := e/(e + 1); d[1] = 1; d[n_] := Denominator[Plus @@ f @@@ FactorInteger[n]]; Select[lps, d[#] == 1 &]

A346017 a(n) is the least number k such that the average number of distinct prime factors of the divisors of k is equal to n.

Original entry on oeis.org

1, 6, 210, 30030, 9699690, 6469693230, 6016814703900, 9127507905816300, 18446693477654742300, 57682810504626379172100, 235749646532408011676372700, 1221890417977470724518639704100, 8011935470678275540668720539783700, 69167038918365552742593064419952682100

Offset: 0

Amiram Eldar, Jul 01 2021

nonn

This sequence is well-defined since for each n there are infinitely many numbers k such that f(k) = A346009(k)/A346010(k) = n. E.g., if k is a squarefree number (A005117) with 2*n prime factors, then f(k) = n.

All the terms are least integers of their prime signature (A025487) since f(n) depends only on the prime signature of n.

Amiram Eldar, Table of n, a(n) for n = 0..16

Cf. A005117, A346009, A346010.

Subsequence of A025487, A346014 and A346015.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; f[p_, e_] := e/(e + 1); f[1] = 0; f[n_] := Plus @@ f @@@ FactorInteger[n]; s = Table[0, {7}]; Do[n = lps[[k]]; f1 = f[n]; If[IntegerQ[f1] && s[[f1+1]] == 0, s[[f1+1]] = n], {k, 1, Length[lps]}]; s

cp's OEIS Frontend

A346015 Least integers of their prime signature (A025487) whose average number of distinct prime factors of their divisors is an integer.

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