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

Author

Amiram Eldar, Mar 05 2025

Keywords

• nonn
• easy

Comments

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.

Links

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

Crossrefs

Cf. A063734.

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

Programs

Mathematica

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

PARI

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

Formula

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