A063734 -id:A063734 - 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)).

A383390 Numbers k such that k^2 and (k+1)^2 are both abundant numbers.

Original entry on oeis.org

104, 495, 584, 735, 944, 1155, 1364, 1484, 2144, 2204, 2415, 2624, 2924, 2925, 3135, 3255, 3794, 3795, 4304, 4484, 4784, 4844, 5264, 5355, 5445, 5564, 5565, 5655, 5775, 5984, 6104, 6764, 7424, 7455, 7664, 7755, 7875, 8084, 8294, 8295, 8414, 8415, 8924, 9009, 9344, 9944, 9975

Offset: 1

Amiram Eldar, Apr 25 2025

nonn

The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 5, 47, 459, 4655, 46733, 460693, 4612685, 46177602, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00461... .

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

Subsequence of A381738.
A383391 and A096399 are subsequences.
Cf. A005101, A063734.

Select[Range[10000], DivisorSigma[-1, #^2] > 2 && DivisorSigma[-1, (#+1)^2] > 2 &]

is1(k) = {my(f = factor(k)); prod(i = 1, #f~, f[i,2] *= 2); sigma(f, -1) > 2;}
list(lim) = {my(q1 = is1(1), q2); for(k = 2, lim, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A383391 Numbers k such that k^2, (k+1)^2 and (k+2)^2 are all abundant numbers.

Original entry on oeis.org

2924, 3794, 5564, 8294, 8414, 10064, 13454, 19304, 22154, 22814, 35684, 39974, 40544, 40754, 41768, 46214, 49994, 52064, 56264, 60884, 63854, 65624, 68354, 68474, 69068, 70244, 78974, 84824, 88604, 92168, 93224, 95354, 100694, 102464, 106028, 107084, 111110, 111824

Offset: 1

Amiram Eldar, Apr 25 2025

nonn

Are there numbers k such that k^2, (k+1)^2, (k+2)^2 and (k+3)^2 are all abundant numbers? There are none below 2.5*10^10.
Are there odd terms in this sequence? There are none below 2.5*10^10.
From David A. Corneth, Apr 26 2025: (Start)
If it exists then it is at least sqrt(A002110(24)/2 * 155925 + 1) - 1 ~= 4.3*10^19.
Proof: Exactly two of k, k+1, k+2 and k+3 are odd. Those two are coprime and differ by 2. Let them be m and m+2. Then sigma(m) > 2*m and sigma(m+2) > 2*(m+2). As they are coprime we have sigma(m*(m+2)) > 2*m*2*(m+2) so for a lower bound we look for the smallest odd t that sigma(t) > 4*t. The partial product of p / (p-1) for odd primes p first exceeds 4 when odd primes <= 79 are multiplied so t is divisible by 3 * 5 * 7 * ... * 79. A small search of multiples of this number gives A002110(24)/2 * 155925.
k * (k + 2) >= A002110(24)/2 * 155925 so k * (k + 2) + 1 = (k + 1)^2 >= A002110(24)/2 * 155925 + 1. Taking square roots on both sides and keeping the positive root gives the desired lower bound. (End)
From Yifan Xie, Apr 30 2025: (Start)
Both types of numbers exist, but the constructed ones are too large to be displayed here. For numbers k such that k^2, (k+1)^2, (k+2)^2 and (k+3)^2 are all abundant numbers, choose 4 disjoint subsets of the primes P_1, P_2, P_3 and P_4, and let the product of elements in P_i divide k+i-1. This is achievable because of the Chinese remainder theorem. If P_i contains p_1, ..., p_k, then sigma((k+i-1)^2)/(k+i-1)^2 >= Product_{i=1..k} (p_i+1)/p_i.
We are able to make the right hand side larger than 2 for each i because the infinite product Product_{p is prime} (p+1)/p = Product_{p is prime} (1+1/p) = Sum_{k is squarefree} 1/k diverges, since the squarefree numbers have asymptotic density 6/Pi^2.
For odd terms in this sequence, we can use only the odd primes to construct 3 prime subsets instead, and add a constraint that k == 1 (mod 2) after which the Chinese remainder theorem still applies. (End)

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

Subsequence of A381738 and A383390.

Cf. A002110, A005101, A063734, A096536.

q[n_] := q[n] = DivisorSigma[-1, n^2] > 2; Select[Range[120000], AllTrue[# + {0, 1, 2}, q] &]

is1(k) = {my(f = factor(k)); prod(i = 1, #f~, f[i,2] *= 2); sigma(f, -1) > 2;}
list(lim) = {my(q1 = is1(1), q2 = is1(2), q3); for(k = 3, lim, q3 = is1(k); if(q1 && q2 && q3, print1(k-2, ", ")); q1 = q2; q2 = q3);}

A380892 Hexagonal numbers that are abundant.

Original entry on oeis.org

66, 120, 276, 378, 630, 780, 1128, 1326, 1540, 1770, 2016, 2556, 2850, 3160, 3486, 3828, 4560, 4950, 5778, 6216, 7140, 7626, 7875, 8646, 9180, 9730, 10296, 10878, 12090, 12720, 14028, 14706, 15400, 16110, 16836, 17955, 18336, 19110, 19900, 20706, 21528, 21945, 23220, 24090, 24976

Offset: 1

Massimo Kofler, Feb 07 2025

nonn

The least term that is coprime to 6 is a(30415179) = 9820742934657655. - Amiram Eldar, Feb 07 2025

			66 = 2*3*11 is a term since it is a hexagonal number and less than the sum of its proper divisors 78.
120 = 2^3*3*5 is a term since it is a hexagonal number and less than the sum of its proper divisors 240.
7875 = 3^2*5^3*7 is a term since it is a hexagonal number and less than the sum of its proper divisors 8349.

Intersection of A000384 and A005101.

Cf. A074315, A063734, A117794, A379264.

Select[Table[n*(2*n-1), {n, 1, 125}], DivisorSigma[-1, #] > 2 &] (* Amiram Eldar, Feb 07 2025 *)

select(x->sigma(x)>2*x, vector(150, k, k*(2*k-1))) \\ Michel Marcus, Feb 07 2025

cp's OEIS Frontend

A381738 Numbers k such that k^2 is abundant.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383390 Numbers k such that k^2 and (k+1)^2 are both abundant numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A383391 Numbers k such that k^2, (k+1)^2 and (k+2)^2 are all abundant numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A380892 Hexagonal numbers that are abundant.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI