A363171 -id:A363171 - cp's OEIS frontend

A363172 Primitive terms of A363171: terms of A363171 with no proper divisor in A363171.

6, 10, 14, 44, 52, 105, 136, 152, 184, 232, 248, 286, 374, 418, 442, 495, 506, 592, 656, 688, 752, 848, 944, 976, 1292, 1564, 1748, 1755, 1972, 2108, 2144, 2145, 2204, 2272, 2336, 2356, 2516, 2528, 2656, 2668, 2788, 2805, 2812, 2848, 2852, 2924, 2925, 3104, 3116

Offset: 1

Amiram Eldar, May 19 2023

nonn

If k is a term then m*k is a term of A363171 for all m >= 1.

The least odd term is a(6) = 105, and the least term that is coprime to 6 is a(34832) = 37182145.

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

Cf. A363171.

q[n_] := DivisorSigma[-1, n * Times @@ FactorInteger[n][[;; , 1]]] > 2; primQ[n_] := q[n] && AllTrue[Divisors[n], # == n || ! q[#] &]; Select[Range[3200], primQ]

A064549(n) = { my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2]+1)); };
isA363171(n) = sigma(A064549(n), -1) > 2;
is(n) = { if(!isA363171(n), return(0)); fordiv(n, d, if(d < n && isA363171(d), return(0))); return(1) };

A381738 Numbers k such that k^2 is abundant.

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, 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)).

A363169 Powerful abundant numbers: numbers that are both powerful (A001694) and abundant (A005101).

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 784, 800, 864, 900, 968, 972, 1000, 1152, 1296, 1352, 1372, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2304, 2500, 2592, 2700, 2704, 2744, 2916, 3136, 3200, 3456, 3528, 3600, 3872, 3888, 4000

Offset: 1

Amiram Eldar, May 19 2023

nonn

The least odd term is a(90) = 11025, and the least term that is coprime to 6 is 1382511906801025.

Are there two consecutive integers in this sequence? There are none below 10^22.

			36 = 2^2 * 3^2 is a term since it is powerful, and sigma(36) = 91 > 2*36 = 72.

Amiram Eldar, Table of n, a(n) for n = 1..10534 (terms not exceeding 10^8)

Intersection of A001694 and A005101.
Cf. A180114, A363170, A363171.
Subsequences: A307959, A328136, A356871.

Select[Range[4000], DivisorSigma[-1, #] > 2 && Min[FactorInteger[#][[;;, 2]]] > 1 &]

is(n) = { my(f = factor(n)); n > 1 && vecmin(f[, 2]) > 1 && sigma(f, -1) > 2; }

cp's OEIS Frontend

A363172 Primitive terms of A363171: terms of A363171 with no proper divisor in A363171.

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A381738 Numbers k such that k^2 is abundant.

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A363169 Powerful abundant numbers: numbers that are both powerful (A001694) and abundant (A005101).

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Mathematica

PARI