A174830 -id:A174830 - cp's OEIS frontend

A156942 Odd abundant numbers whose abundance is odd.

11025, 99225, 245025, 275625, 342225, 540225, 893025, 1334025, 1863225, 2205225, 2480625, 3080025, 3186225, 3980025, 4601025, 4862025, 5832225, 6125625, 6890625, 7868025, 8037225, 8555625, 9272025, 9828225, 10595025, 10989225

Offset: 1

Robert G. Wilson v, Feb 18 2009

nonn

Number of terms <10^n: 0, 0, 0, 0, 2, 7, 24, 83, 250, 792, 2484, 7988, 25383, 80082, ..., . Not all are a multiple of 25, i.e.; 81162081 = 9009^2 = (9*7*11*13)^2. See A156943.
Any term must be an odd square. Square roots are in A174830.
Indeed, the sum of divisors of any number isn't odd unless it's a square or twice a square (A028982), and to get the abundance, twice the number is subtracted, so the parity remains the same. - M. F. Hasler, Jan 26 2020
Question: Is this a subsequence of A379503? (Is A379504(a(n)) > 0 for all n? See A379951). The first 15000 terms are all included there. - Amiram Eldar and Antti Karttunen, Jan 06 2025
Question 2: Is A379505(a(n)) > 1 for all n, especially if there are no quasiperfect numbers (numbers k such that sigma(k) = 2k+1)? - Antti Karttunen, Jan 06 2025
From Amiram Eldar, Jan 16 2025: (Start)
The least term that is not divisible by 5 is a(75) = 81162081.
The least term that is not divisible by 3 is a(296889) = 1382511906801025.
The least term that is coprime to 15 is 15285071557677427358507559514565648611799881. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)

Cf. A005101, A005231, A028982, A033880, A156903, A174830, A379503, A379504, A379505, A379951 [= A379504(a(n))].

Subsequences: A156943, A325311 (thus also A379490), A347890, A379949 (terms that are primitive abundant).

fQ[n_] := Block[{ds = DivisorSigma[1, n] - 2 n}, ds > 0 && OddQ@ ds]; Select[ Range[1, 12006223, 2], fQ @# &]

is(n)=my(s=sigma(n)); n%2 && s>2*n && (s-2*n)%2 \\ Charles R Greathouse IV, Feb 21 2017

a(n) = A174830(n)^2. - M. F. Hasler, Jan 26 2020

Edited by Robert G. Wilson v at the suggestion of T. D. Noe, Mar 30 2010

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)).

A325312 Odd numbers k for which sigma(k^2) == 3 (mod 4) and sigma(k^2) > 2*k^2.

Original entry on oeis.org

105, 315, 495, 735, 945, 1155, 1485, 1995, 2205, 2415, 2625, 2835, 2925, 3135, 3255, 3315, 3465, 3795, 4455, 4515, 4935, 5115, 5145, 5445, 5655, 5985, 6195, 6615, 6825, 7035, 7095, 7245, 7455, 7605, 7755, 7875, 8085, 8295, 8505, 8715, 8775, 8925, 9009, 9405, 9735, 9765, 9945, 10395, 10725, 10815, 11235, 11385, 11781, 12375

Offset: 1

Antti Karttunen, Apr 20 2019

nonn

Square roots of A325311, odd abundant numbers k for which sigma(k) == 3 mod 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000196, A325311.

Intersection of A174830 and A324909.

Select[Range[1, 12375 , 2], Mod[(s = DivisorSigma[1, #^2]), 4] == 3 && s > 2*#^2 &] (* Amiram Eldar, Apr 05 2024 *)

isA325312(n) = { my(t=sigma(n*n)); (n%2 && 3==(t%4) && t>(2*n*n)); };

a(n) = sqrt(A325311(n)).

A379950 Numbers k such that k^2 is an odd primitive abundant number (A006038).

Original entry on oeis.org

585, 32085, 41925, 46665, 121605, 134589, 181305, 212175, 388455, 495465, 544065, 839865, 1061565, 1152921, 1165515, 1243275, 1247103, 1335411, 1676829, 2151075, 2290869, 2478075, 2771835, 2838165, 3016725, 3122847, 3156795, 4571415, 4738041, 5153841, 5558985, 6125049, 7471425, 7676775, 7780101, 7822425, 8259867

Offset: 1

Antti Karttunen, Jan 07 2025

nonn

Amiram Eldar, Table of n, a(n) for n = 1..3500 (terms 1..725 from Antti Karttunen)

Square roots of A379949.
Subsequence of A174830.
Cf. A000196, A006038.

isok(k) = if(!(k % 2), 0, my(f = factor(k)); for(i = 1, #f~, f[i, 2] *= 2); if(sigma(f, -1) <= 2, return(0)); for(i = 1, #f~, f[i, 2] -= 1; if(sigma(f, -1) > 2, return(0)); f[i, 2] += 1); 1); \\ Amiram Eldar, Mar 12 2025

a(n) = A000196(A379949(n)).

A179271 Odd long legs `B` of more than one primitive Pythagorean triangle.

Original entry on oeis.org

2145, 3315, 3465, 4095, 4845, 5005, 5865, 6435, 6545, 6555, 7735, 8645, 9009, 9945, 10005, 10695, 11305, 11781, 13167, 13485, 13685, 13923, 14535, 15015, 15295, 15561, 16065, 16095, 17017, 17205, 17255, 17835, 17955, 18837, 19019, 19065

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 06 2010

nonn

2145,752,2273;2145,1568,2657;;3315,812,3413;3315,2852,4373;

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A024407, A024408, A024409, A024410, A024411, A161962, A174830

lst1={};lst={0};Do[Do[If[GCD[a,b]==1,c=Sqrt[a^2+b^2];If[IntegerQ[c],AppendTo[lst,b];L=Length[lst];If[lst[[L]]==lst[[L-1]]&&OddQ[lst[[L]]],Print[lst[[L]]];AppendTo[lst1,lst[[L]]]]]],{a,b-1,3,-1}],{b,4,4*7!}];lst1

a(20) - a(36) Robert G. Wilson v, Jul 12 2010

A347891 Odd numbers k such that sigma(k^2) > 2*k^2 and A003415(sigma(k^2)) < k^2.

Original entry on oeis.org

495, 735, 945, 1485, 1755, 2205, 2475, 2625, 2805, 3315, 3675, 3795, 4455, 4785, 4845, 5145, 5265, 5445, 6615, 6765, 7425, 7605, 7755, 8085, 8415, 8505, 8745, 8775, 9735, 11025, 12375, 12495, 13365, 13965, 14025, 15435, 15795, 16065, 16335, 16905, 17595, 18375, 19845, 20295, 21315, 22185, 22275, 22785, 22815, 23265

Offset: 1

Antti Karttunen, Sep 19 2021

nonn

Odd numbers whose square is abundant and present in A343216.

The first term that is not a multiple of 5 is a(146) = 82467.

Index entries for sequences related to sigma(n)

Square roots of A347890. Subsequence of A174830.

Cf. A000196, A000203, A003415, A005231, A343216.

ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 24000, 2], DivisorSigma[1, #^2] > 2*#^2 && ad[DivisorSigma[1, #^2]] < #^2 &] (* Amiram Eldar, Sep 19 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347891(n) = if(!(n%2),0,my(u=n*n); (A003415(sigma(u))(2*u)));

a(n) = A000196(A347890(n)).

cp's OEIS Frontend

A156942 Odd abundant numbers whose abundance is odd.

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A381738 Numbers k such that k^2 is abundant.

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A325312 Odd numbers k for which sigma(k^2) == 3 (mod 4) and sigma(k^2) > 2*k^2.

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A379950 Numbers k such that k^2 is an odd primitive abundant number (A006038).

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A179271 Odd long legs `B` of more than one primitive Pythagorean triangle.

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A347891 Odd numbers k such that sigma(k^2) > 2*k^2 and A003415(sigma(k^2)) < k^2.

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