A228058 -id:A228058 - cp's OEIS frontend

A331751 Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

2, 6, 27, 28, 84, 270, 496, 1053, 1120, 1488, 1625, 1638, 3360, 3780, 4875, 8128, 10530, 24384, 66960, 147420, 167400, 406224, 611226, 775000, 872960, 943250, 1097280, 1245699, 1255338, 1303533, 1464320, 1686400, 1740024, 1922375, 1952500, 2011625, 2193408, 2325000, 2611440, 2618880, 2829750, 2941029, 4392960

Offset: 1

Antti Karttunen, Feb 05 2020

nonn

Numbers k such that A097248(sigma(k)) is equal to A097248(2*k).
Numbers k such that A331750(k) is equal to 1+A048675(k), which in turn is equal to A048675(A225546(2*k)) = A048675(2*A225546(k)).
Among the first 60 terms, 15 are odd: 27, 1053, 1625, 4875, 1245699, 1303533, 1922375, 2011625, 2941029, 5767125, 6034875, 12733875, 17137575, 26316675, 29362905, and only 1053 = 3^4 * 13 is in A228058.
Note that the condition A090880(sigma(k)) == A090880(2*k) appears to be much more constrained.

			For n = 1053 = 3^4 * 13^1, A331750(1053) = A331750(81) + A331750(13) = 32+9 = 41, while A048675(2*1053) = A048675(2)+A048675(81)+A048675(13) = 1+8+32 = 41 also, thus 1053 is included in this sequence.
For n = 3360 = 2^5 * 3^1 * 5^1 * 7^1, A331750(3360) = A331750(32)+A331750(3)+A331750(5)+A331750(7) = 12+2+3+3 = 20, while A048675(2*3360) = A048675(2)+A048675(32)+A048675(3)+A048675(5)+A048675(7) = 1+5+2+4+8 = 20 also, thus 3360 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..60 (up to the term 33550336).
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A048675, A090880, A097246, A097248, A331750, A331752, A332208.

Cf. A000396 (a subsequence).

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i, 1]+1)^(f[i, 2]\2))*((f[i, 1])^(f[i, 2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
isA331751(n) = (A097248(2*n)==A097248(sigma(n)));

A065235 Odd numbers which can be written in precisely one way as sum of a subset of their proper divisors.

Original entry on oeis.org

8925, 32445, 351351, 442365, 159427275, 159587925, 159677175, 159784275, 159837825, 159855675, 159944925, 159962775, 160016325, 160105575, 160266225, 160284075, 160391175, 160444725, 160480425, 160533975, 160551825, 160766025, 161015925, 161033775, 161069475

Offset: 1

Jud McCranie, Oct 23 2001

nonn

From Antti Karttunen, Nov 28 2024: (Start)
Characteristic function of this sequence is c(n) = A000035(n)*A378448(n).
The only non-multiples of 25 among the first 10000 terms are a(2)..(4): 32445 = 3^2 * 5 * 7 * 103, 351351 = 3^3 * 7 * 11 * 13^2 and 442365 = 3 * 5 * 7 * 11 * 383, while the other 9997 terms are all of the form 25 * some squarefree number. No terms of A228058 occur among the initial 10000 terms. Compare also to A348743.
(End)

			See A064771 for an example when the number is even.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Index entries for sequences where any odd perfect numbers must occur

Odd terms in A064771 (a unique subset of proper divisors sums to the number).

Cf. A000035, A033630, A065205, A136446, A228058, A348743, A378448.

{k such that k is odd and A065205(k) = 1}. - Antti Karttunen, Nov 28 2024

Definition clarified by M. F. Hasler, Apr 08 2008

More terms from Giovanni Resta, Oct 04 2019

A325807 Number of ways to partition the divisors of n into complementary subsets x and y for which gcd(n-Sum(x), n-Sum(y)) = 1. (Here only distinct unordered pairs of such subsets are counted.)

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 1, 8, 3, 4, 1, 16, 1, 4, 2, 16, 1, 16, 1, 16, 4, 4, 1, 40, 3, 3, 4, 1, 1, 40, 1, 32, 2, 4, 4, 244, 1, 4, 4, 48, 1, 40, 1, 16, 8, 3, 1, 220, 3, 27, 2, 10, 1, 32, 4, 64, 4, 4, 1, 672, 1, 4, 14, 64, 4, 40, 1, 13, 2, 64, 1, 1205, 1, 4, 16, 10, 4, 40, 1, 236, 15, 4, 1, 864, 4, 3, 2, 64, 1, 640, 2, 16, 4, 4, 2, 537, 1, 26, 8, 241, 1, 40, 1, 64, 40

Offset: 1

Antti Karttunen, May 24 2019

nonn

			For n = 1, its divisor set [1] can be partitioned only to an empty set [] and set [1], with sums 0 and 1 respectively, and gcd(1-0,1-1) = gcd(1,0) = 1, thus this partitioning is included, and a(1) = 1.
For n = 3, its divisor set [1, 3] can be partitioned as [] and [1,3] (sums 0 and 4, thus gcd(3-0,3-4) = 1), [1] and [3] (sums 1 and 3, thus gcd(3-1,3-3) = 2), thus a(3) = 1, and similarly a(p) = 1 for any other odd prime p as well.
For n = 6, its divisor set [1, 2, 3, 6] can be partitioned in eight ways as:
  [] and [1, 2, 3, 6] (sums 0 and 12, gcd(6-0, 6-12) = 6),
  [1, 2] and [3, 6]   (sums 3 and 9,  gcd(6-3, 6-9) = 3),
  [1, 3] and [2, 6]   (sums 4 and 8,  gcd(6-4, 6-8) = 2),
  [2] and [1, 3, 6]   (sums 2 and 10, gcd(6-2, 6-10) = 4),
  [3] and [1, 2, 6]   (sums 3 and 9,  gcd(6-3, 6-9) = 3),
  [6] and [1, 2, 3]   (sums 6 and 6,  gcd(6-6, 6-6) = 0),
  [1] and [2, 3, 6]   (sums 1 and 11, gcd(6-1, 6-11) = 5),
  [1, 6] and [2, 3]   (sums 7 and 5,  gcd(6-7, 6-5) = 1),
with only the last partitioning satisfying the required condition, thus a(6) = 1.
For n = 10, its divisor set [1, 2, 5, 10] can be partitioned in eight ways as:
  [] and [1, 2, 5, 10] (sums 0 and 18, gcd(10-0, 10-18) = 2),
  [1, 2] and [5, 10]   (sums 3 and 15, gcd(10-3, 10-15) = 1),
  [1, 5] and [2, 10]   (sums 6 and 12, gcd(10-6, 10-12) = 2),
  [2] and [1, 5, 10]   (sums 2 and 16, gcd(10-2, 10-16) = 2),
  [5] and [1, 2, 10]   (sums 5 and 13, gcd(10-5, 10-13) = 1),
  [10] and [1, 2, 5]   (sums 10 and 8, gcd(10-10, 10-8) = 2),
  [1] and [2, 5, 10]   (sums 1 and 17, gcd(10-1, 10-17) = 1),
  [1, 10] and [2, 5]   (sums 11 and 7, gcd(10-11, 10-7) = 1),
of which four satisfy the required condition, thus a(10) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..1079

Cf. A000203, A000396, A065091, A100577, A324213, A325806, A325809.

Table[Function[d, Count[DeleteDuplicates[Sort /@ Map[{#, Complement[d, #]} &, Subsets@ d]], ?(CoprimeQ @@ (n - Total /@ #) &)]]@ Divisors@ n, {n, 105}] (* _Michael De Vlieger, May 27 2019 *)

A325807(n) = { my(divs=divisors(n), s=sigma(n),r); sum(b=0,(2^(-1+length(divs)))-1,r=sumbybits(divs,2*b);(1==gcd(n-(s-r),n-r))); };
sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); };

For all n >= 1:
a(n) <= A100577(n).
a(A065091(n)) = 1, a(A000396(n)) = 1.
a(A228058(n)) = A325809(n).

A348743 Odd nonsquares k for which A161942(k) >= k, where A161942 is the odd part of sigma.

Original entry on oeis.org

2205, 19845, 108045, 143325, 178605, 187425, 236925, 266805, 319725, 353925, 372645, 407925, 452025, 462825, 584325, 637245, 646425, 658125, 672525, 789525, 796005, 804825, 845325, 920205, 972405, 981225, 1007325, 1055925, 1069425, 1102725, 1113525, 1116225, 1166445, 1201725, 1245825, 1289925, 1378125, 1380825, 1442925

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

The first non-multiples of 5 are a(103) = 6243237 and a(125) = 8164233.
From Antti Karttunen, Nov 28 2024: (Start)
This is not a subsequence of A228058. At least k = A000040(28)*(A002110(27)/2)^2 = 15388519572341080054329140040512468358441210638435506649120749687401476705908239675 is a number of the form 4m+3 such that A161942(k) >= k.
Another such number is A000040(28)*81*(A002110(25)/6)^2 = 1279741205456530915782536871495922949062895982530933679752838870798129159675.
Question: What is the smallest term of this sequence that is of the form 4m+3, and thus not in A386427 (in A191218 and in A228058)?
(End)

Intersection of A088828 and A348742.
Cf. A386427 (a subsequence, which agrees for a very long time).
Cf. A033630, A065205, A083206, A161942, A191218, A228058, A336702, A348741.
Cf. also A065235, A162284.

A000265(n) = (n >> valuation(n, 2));
isA348743(n) = ((n%2)&&!issquare(n)&&A000265(sigma(n))>=n); \\ Edited Nov 28 2024

Definition changed (from > to >=) to formally include also any hypothetical odd perfect numbers - Antti Karttunen, Nov 28 2024

Comment removed, because it was more related to sequence A386427. - Antti Karttunen, Aug 21 2025

A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Original entry on oeis.org

117, 775, 10309, 56347, 88723, 2896363, 9597529, 12326221, 12654079, 25774633, 29817121, 63455131, 105100903, 203822581, 261019543, 296765173, 422857021, 573332713, 782481673, 900952687, 1129152721, 3350861677, 3703086229, 7395290407, 9347001661, 9350506057

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Nineteen initial terms factored:
   n       a(n)   factorization   A060681(a(n))/A318505(a(n))
      117 =   3^2 *    13,           (3)
      775 =   5^2 *    31,          (10)
    10309 =  13^2 *    61,          (39)
    56347 =  29^2 *    67,          (58)
    88723 =  17^2 *   307,         (136)
  2896363 =  41^2 *  1723,         (820)
  9597529 =  73^2 *  1801,        (1314)
 12326221 =  59^2 *  3541,        (1711)
 12654079 = 113^2 *   991,         (904)
 25774633 =  71^2 *  5113,        (2485)
 29817121 =  97^2 *  3169,        (2328)
 63455131 =  89^2 *  8011,        (3916)
105100903 = 101^2 * 10303,        (5050)
203822581 = 157^2 *  8269,        (6123)
261019543 = 349^2 *  2143,        (2094)
296765173 = 131^2 * 17293,        (8515)
422857021 = 233^2 *  7789,        (6757)
573332713 = 331^2 *  5233,        (4965)
782481673 = 167^2 * 28057,       (13861).
Note how the quotient (in the rightmost column) seems always to be a multiple of non-unitary prime factor and less than the unitary prime factor.
For p, q prime, if p^2+p+1 = kq and k+1|p-1, then p^2*q is in this sequence. - Charlie Neder, Jun 09 2019

Subsequence of A326063.
Cf. A032742, A060681, A246282, A318505.
Cf. also A228058, A325981, A326131, A326141.

Select[Range[15, 10^6 + 1, 2], And[! PrimePowerQ@ #1, Mod[#1 - #2, #2 - #3] == 0] & @@ {#, DivisorSigma[1, #] - #, Divisors[#][[-2]]} &] (* Michael De Vlieger, Jun 22 2019 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
isA326064(n) = if((n%2)&&(2!=isprimepower(n)), my(s=A032742(n), t=sigma(n)-s); (gcd(t-n, n-A032742(n)) == t-n), 0);

More terms from Amiram Eldar, Dec 24 2020

A332228 Odd numbers n, not powers of primes, such that sigma(n) is congruent to 2 modulo 8.

Original entry on oeis.org

153, 325, 369, 657, 725, 801, 833, 845, 873, 925, 1017, 1233, 1325, 1377, 1445, 1525, 1737, 2009, 2057, 2097, 2169, 2313, 2525, 2529, 2725, 2817, 2925, 3033, 3177, 3321, 3577, 3609, 3681, 3725, 3757, 3897, 3925, 4041, 4113, 4205, 4325, 4361, 4525, 4689, 4753, 4901, 4925, 4961, 5121, 5193, 5337, 5409, 5537, 5553, 5725

Offset: 1

Antti Karttunen, Feb 13 2020

nonn

Proof that any odd perfect number, if such numbers exist at all, has to reside in this sequence: As all terms in A228058 are = 1 modulo 4 (their binary expansion ends as "01"), and taking sigma of an odd perfect number would multiply it by two (shift one bit-position left), the base-2 expansion of that result would end as "010", i.e., sigma(k) modulo 8 should be 2 (not 6) for such numbers k.

Cf. A000203, A332229.

Subsequence of A228058, of A332226 and of A332227.

isA332228(n) = ((n%2)&&!isprimepower(n)&&2==(sigma(n)%8));

A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

Original entry on oeis.org

153, 801, 1773, 3725, 4689, 4753, 5013, 6957, 8577, 8725, 9549, 9873, 11493, 13437, 14409, 15381, 18621, 19269, 21213, 21537, 23481, 25101, 26073, 26225, 28989, 29161, 29313, 29961, 32229, 33849, 34173, 36117, 38061, 39033, 40653, 42597, 43893, 47457, 47781, 48725, 48753, 51669, 52317, 54261, 56953, 57177, 57501

Offset: 1

Antti Karttunen, Aug 17 2025

nonn

By definition, the sequence contains all odd perfect numbers, and also includes any hypothetical odd triperfect number that is not a multiple of 3 (see A005820 and A347391), and similarly, any odd term of A046060 that is not a multiple of 5, etc. If there are no squares in this sequence (see conjecture in A386424), then the latter categories of numbers certainly do not exist, and this is then a subsequence of A228058.

The first nondeficient term is a(32315) = 81022725. See A386426.

Intersection of A071904 and A386424.
Nonsquare terms form a subsequence of A228058.
Cf. A000203, A003557, A057521, A386426 (nondeficient terms).
Cf. also A324647, A349749.

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[9,57501,2],!PrimeQ[#]&&a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386425(n) = ((n>1) && (n%2) && !isprime(n) && (A057521(sigma(n))==A057521(n)));

{k | k is odd composite and A003557(A000203(k)) = A003557(k)}.

A162284 Odd numbers n such that b(n) >= n and b(b(n)) >= n, where b(n) = A161942(n) = oddpart(sigma(n)).

Original entry on oeis.org

1, 81, 18966025, 135187129, 164275489, 350561925, 445421025, 598047025, 649587169, 748733769, 850830561, 960362325, 1055925025, 1341097641, 1406175001, 1476326929, 1520766009, 1536248025, 1591004025, 1735566525

Offset: 1

Franklin T. Adams-Watters, Jun 29 2009

nonn

Applying b a third time produces a smaller value for every number in this sequence up to 10^10, except 1.
Of the first 43 terms, only the following nine are not squares: 350561925, 960362325, 1591004025, 1735566525, 1753206525, 1831175325, 4558583925, 6745097205, 8766517725, and incidentally, all of them are terms of A228058. - Antti Karttunen, Jun 16 2019
But see also comments in A348743. - Antti Karttunen, Nov 30 2024

Cf. A161942, A228058, A326042.
Subsequence of A348742.
Cf. also A348743.

Values for n > 3 from Robert Gerbicz

A228056 Numbers of the form p * m^2, where p is prime and m > 1.

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 32, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 125, 128, 147, 148, 153, 162, 164, 171, 172, 175, 176, 180, 188, 192, 200, 207, 208, 212, 236, 242, 243, 244, 245, 252, 261, 268, 272, 275, 279, 284

Offset: 1

T. D. Noe, Aug 13 2013

nonn

This sequence is the first step toward candidates for odd perfect numbers, A228058.

T. D. Noe, Table of n, a(n) for n = 1..10000
Raghavendra Bhat, Distribution of Square Prime Numbers, arXiv:2109.10238 [math.NT], 2021.
Raghavendra Bhat, An Algebraic Structure for Square-Prime Numbers, arXiv:2303.14296 [math.GM], 2023.
Raghavendra Bhat, Cristian Cobeli, and Alexandru Zaharescu, Filtered rays over iterated absolute differences on layers of integers, arXiv:2309.03922 [math.NT], 2023. See 3.1 p. 9.

Cf. A228057, A228058.

Cf. A124010.

import Data.List (partition)
a228056 n = a228056_list !! (n-1)
a228056_list = filter f [1..] where
   f x = length us == 1 && (head us > 1 || not (null vs)) where
         (us,vs) = partition odd $ a124010_row x
-- Reinhard Zumkeller, Aug 14 2013

nn = 300; Union[Select[Flatten[Table[p*n^2, {p, Prime[Range[PrimePi[nn/4]]]}, {n, 2, Sqrt[nn/2]}]], # < nn &]]

list(lim)=my(v=List()); forfactored(n=2, lim\1, my(e=n[2][, 2]); if(vecsum(e%2)==1 && e!=[1]~, listput(v, n[1]))); Vec(v); \\ Charles R Greathouse IV, Oct 01 2021

from math import isqrt
from sympy import primepi
def A228056(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//y**2) for y in range(2,isqrt(x)+1))
    return bisection(f,n,n) # Chai Wah Wu, Jun 06 2025

Bhat proves there are ~ (Pi^2/6-1)*x/log x members of this sequence up to x, so a(n) ~ kn log n with k = 6/(Pi^2-6) = 1.550546.... - Charles R Greathouse IV, Oct 01 2021

A324898 Odd numbers k such that sigma(k) is congruent to 2 modulo 4 and k = A318458(k), where A318458(k) is bitwise-AND of k and sigma(k)-k.

Original entry on oeis.org

236925, 3847725, 51122925, 69468525, 151141725, 154669725, 269748225, 344211525, 415565325, 445817925, 551569725, 1111904325, 1112565825, 1113756525, 1175717025, 1400045625, 1631666925, 1695170925, 1820873925, 1915847325, 1946981925, 2179080225, 2321121825, 2453690925, 2460041325, 2491740225, 3223500525, 3493517445, 3775103325

Offset: 1

Antti Karttunen, Apr 19 2019

nonn

If this sequence has no common terms with A324647, or no terms common with A324727, then there are no odd perfect numbers.
The first 29 terms factored:
      236925 = 3^6 * 5^2 *  13,
     3847725 = 3^2 * 5^2 *  7^2 * 349,
    51122925 = 3^2 * 5^2 *  7^2 * 4637,
    69468525 = 3^2 * 5^2 *  7^2 * 6301,
   151141725 = 3^2 * 5^2 *  7^2 * 13709,
   154669725 = 3^2 * 5^2 *  7^2 * 14029,
   269748225 = 3^6 * 5^2 * 19^2 * 41,
   344211525 = 3^4 * 5^2 *  7^2 * 3469,
   415565325 = 3^2 * 5^2 *  7^2 * 37693,
   445817925 = 3^4 * 5^2 *  7^2 * 4493,
   551569725 = 3^2 * 5^2 *  7^4 * 1021,
  1111904325 = 3^2 * 5^2 *  7^2 * 100853,
  1112565825 = 3^2 * 5^2 *  7^2 * 100913,
  1113756525 = 3^2 * 5^2 *  7^2 * 101021,
  1175717025 = 3^4 * 5^2 *  7^2 * 17^2 * 41,
  1400045625 = 3^2 * 5^4 * 11^4 * 17,
  1631666925 = 3^2 * 5^2 *  7^2 * 147997,
  1695170925 = 3^2 * 5^2 *  7^2 * 153757,
  1820873925 = 3^4 * 5^2 *   13 * 263^2, [Here the unitary prime is not the largest]
  1915847325 = 3^2 * 5^2 *  7^2 * 173773,
  1946981925 = 3^2 * 5^2 *  7^2 * 176597,
  2179080225 = 3^4 * 5^2 *  7^2 * 21961,
  2321121825 = 3^4 * 5^2 * 11^2 * 9473,
  2453690925 = 3^2 * 5^2 *  7^2 * 222557,
  2460041325 = 3^2 * 5^2 *  7^2 * 223133,
  2491740225 = 3^6 * 5^2 * 13^2 * 809,
  3223500525 = 3^2 * 5^2 *  7^2 * 292381,
  3493517445 = 3^6 * 5^1 * 11^2 * 89^2,  [Here the unitary prime is not the largest]
  3775103325 = 3^2 * 5^2 *  7^2 * 342413.
Subsequence of A228058 provided this sequence does not contain any prime powers. - Antti Karttunen, Jun 17 2019
Sequence contains no prime powers up to 10^20. I believe any prime powers must be of the form (4k+1)^(4e+1), in which case I have verified this up to 10^50. - Charles R Greathouse IV, Dec 08 2021

Intersection of A191218 and A324897, also intersection of A191218 and A324649.

Cf. A228058, A318458, A324647, A324727.

Select[Range[10^5, 10^8, 2], And[Mod[#2, 4] == 2, BitAnd[#1, #2 - #1] == #1] & @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Jun 22 2019 *)

for(n=1, oo, if((n%2)&&2==((t=sigma(n))%4)&&(bitand(n, t-n)==n), print1(n,", ")));

cp's OEIS Frontend

A331751 Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

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A065235 Odd numbers which can be written in precisely one way as sum of a subset of their proper divisors.

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A325807 Number of ways to partition the divisors of n into complementary subsets x and y for which gcd(n-Sum(x), n-Sum(y)) = 1. (Here only distinct unordered pairs of such subsets are counted.)

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A348743 Odd nonsquares k for which A161942(k) >= k, where A161942 is the odd part of sigma.

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A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

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A332228 Odd numbers n, not powers of primes, such that sigma(n) is congruent to 2 modulo 8.

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A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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A162284 Odd numbers n such that b(n) >= n and b(b(n)) >= n, where b(n) = A161942(n) = oddpart(sigma(n)).

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