A381741 -id:A381741 - 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)).

A381739 Number k such that k^2 is abundant, and d^2 is nonabundant for any proper divisor d of k.

Original entry on oeis.org

6, 10, 14, 44, 52, 68, 76, 92, 105, 116, 124, 286, 296, 328, 344, 374, 376, 418, 424, 442, 472, 488, 495, 506, 536, 568, 584, 585, 632, 664, 712, 776, 808, 824, 856, 872, 904, 1016, 2096, 2145, 2192, 2224, 2384, 2416, 2512, 2608, 2672, 2768, 2805, 2864, 2896, 3056

Offset: 1

Amiram Eldar, Mar 05 2025

The primitive terms of A381738. Each term of A381738 is a multiple of a term in this sequence.

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

Subsequence of A381738.
A381741 is a subsequence.
Cf. A005101, A263837.

q[k_] := DivisorSigma[-1, k^2] > 2 && AllTrue[Divisors[k], DivisorSigma[-1, #^2] <= 2 || # == k &]; Select[Range[3200], q]

is1(k) = {my(f = factor(k)); prod(i = 1, #f~, f[i,2] *= 2); sigma(f, -1) > 2;}
isok(k) = if(!is1(k), 0, fordiv(k, d, if(d < k && is1(d), return(0))); 1);

A381740 Squarefree numbers k such that k^2 is abundant.

Original entry on oeis.org

6, 10, 14, 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 170, 174, 182, 186, 190, 210, 222, 230, 238, 246, 258, 266, 282, 286, 290, 310, 318, 322, 330, 354, 366, 370, 374, 390, 402, 406, 410, 418, 426, 430, 434, 438, 442, 462, 470, 474, 498, 506, 510

Offset: 1

Amiram Eldar, Mar 05 2025

The least odd term is a(105) = 105.
The least term that is coprime to 6 is a(3637276) = 37182145.
If k is a squarefree number that is divisible by 6, 10 or 14, then it is a term. Therefore a lower bound for the asymptotic density of this sequence is 29/(192*zeta(2)) = 0.0918... .
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 2, 8, 99, 972, 9826, 97610, 979190, 9770801, 97650638, 976893969, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0976... .
If k is a term then any multiple of k that is squarefree is a term. The primitive terms are in A381741.

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

Intersection of A005117 and A381738.
Subsequences: A087248, A381741.
Cf. A013661.

Select[Range[600], SquareFreeQ[#] && DivisorSigma[-1, #^2] > 2 &]

isok(k) = {my(f = factor(k)); if(!issquarefree(f), 0, prod(i = 1, #f~, f[i,2] *= 2); sigma(f, -1) > 2);}

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A381738 Numbers k such that k^2 is abundant.

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A381739 Number k such that k^2 is abundant, and d^2 is nonabundant for any proper divisor d of k.

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A381740 Squarefree numbers k such that k^2 is abundant.

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