A334166 -id:A334166 - cp's OEIS frontend

A381738 Numbers k such that k^2 is abundant.

6, 10, 12, 14, 18, 20, 24, 28, 30, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 116, 120, 124, 126, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174

Offset: 1

Amiram Eldar, Mar 05 2025

First differs from its subsequence A363171 at n = 21: a(21) = 68 is not a term of A363171.
First differs from its subsequence A334166 at n = 204: a(204) = 585 is not a term of A334166.
A334166 is a subsequence because if k is in A334166, then there is a divisor d of k such that d*k is a Zumkeller number, so d*k is abundant (because all the Zumkeller numbers are abundant), and since d*k is a divisor of k^2 then k^2 is also abundant.
Equivalently, numbers k such that d*k is abundant for at least one divisor d of k.
The least odd term is a(36) = 105.
The least term that is coprime to 6 is a(12519603) = 37182145.
If k is divisible by 6, 10 or 14, then it is a term. Therefore a lower bound for the asymptotic density of this sequence is 19/70 = 0.271... .
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 2, 33, 347, 3403, 33728, 336599, 3368889, 33628998, 336480309, 3365049432, ... . Apparently, the asymptotic density of this sequence exists and equals 0.336... .
If k is a term then any positive multiple of k is a term. The primitive terms are in A381739.

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

Cf. A063734.

Subsequences: A005101, A174830, A334166, A363171, A381739, A381740, A381741, A381742.

Select[Range[200], DivisorSigma[-1, #^2] > 2 &]

isok(k) = {my(f = factor(k)); prod(i = 1, #f~, f[i,2] *= 2); sigma(f, -1) > 2;}

a(n) = sqrt(A063734(n)).

A363171 Numbers k such that A064549(k) is an abundant number (A005101).

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 24, 28, 30, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174, 176, 180, 182, 184, 186

Offset: 1

Amiram Eldar, May 19 2023

nonn

First differs from A334166 at n = 21.
The least odd term is a(33) = 105, and the least term that is coprime to 6 is a(11850456) = 37182145.
The ordered values of A003557(A363169(n)): m is a powerful abundant number (A363169) if and only if A003557(m) is in this sequence.
If k is a term then any positive multiple of k is also a term. The primitive terms are in A363172.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 2, 30, 322, 3201, 31863, 318336, 3188014, 31855257, 318427893, 3184885813, 31853300276, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3185... .

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

Cf. A003557, A064549, A334166.

Subsequences: A005101, A363169, A363172.

q[n_] := DivisorSigma[-1, n * Times @@ FactorInteger[n][[;; , 1]]] > 2; Select[Range[200], q]

A064549(n) = { my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2]+1)); };
is(n) = sigma(A064549(n), -1) > 2;

cp's OEIS Frontend

A381738 Numbers k such that k^2 is abundant.

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