A028983 -id:A028983 - cp's OEIS frontend

A152678 Even members of A000203.

4, 6, 12, 8, 18, 12, 28, 14, 24, 24, 18, 20, 42, 32, 36, 24, 60, 42, 40, 56, 30, 72, 32, 48, 54, 48, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 84, 144, 68, 126, 96, 144

Offset: 1

Omar E. Pol, Dec 10 2008

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

Cf. A000203, A028983, A152677, A152679.

Select[DivisorSigma[1,Range[100]],EvenQ] (* Harvey P. Dale, Jun 18 2017 *)

list(lim) = select(x -> !(x % 2), vector(lim, i, sigma(i))); \\ Amiram Eldar, Dec 26 2024

a(n) = sigma(A028983(n)) = A000203(A028983(n)). - Jaroslav Krizek, Oct 06 2009

A371418 The largest aliquot divisor of the sum of divisors of n; a(1) = 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 4, 5, 1, 9, 6, 14, 7, 12, 12, 1, 9, 13, 10, 21, 16, 18, 12, 30, 1, 21, 20, 28, 15, 36, 16, 21, 24, 27, 24, 13, 19, 30, 28, 45, 21, 48, 22, 42, 39, 36, 24, 62, 19, 31, 36, 49, 27, 60, 36, 60, 40, 45, 30, 84, 31, 48, 52, 1, 42, 72, 34, 63, 48, 72

Offset: 1

Amiram Eldar, Mar 23 2024

Carmichael (1921) defined this arithmetic function for the purpose of studying periodic chains that are formed by repeatedly applying the mapping x -> a(x) starting at a given positive integer. This results in a sequence that is analogous to an aliquot sequence.
Periodic chains of cycle 1 are the fixed points of this sequence. 1 and the even perfect numbers (the even terms of A000396) are fixed points. Are there any other numbers k such that a(k) = k?
If a(k) = k and k is even, then a(k) is even and so is sigma(k), and therefore sigma(k) = 2*k and k is an even perfect number. If k is odd, then it is an odd multiperfect number, and no odd multiperfect number above 1 is known.
More specifically, if a(k) = k and k is odd, then k must be a square, and an m-multiperfect number (number k such that sigma(k) = m * k), with m being an odd prime number that is the least prime factor of sigma(k). For example, if there is an odd triperfect number (A005820) then it is a fixed point of this sequence.
Periodic chains of cycle 2 are amicable pairs (A371419 and A371420). Are there any longer cycles?

			The sum of the divisors of 3 is 1 + 3 = 4. The divisors of 4 are 1, 2, 4. 2 is the largest aliquot divisor of 4. Therefore a(3) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.
Eric Weisstein's World of Mathematics, Aliquot Sequence.
Eric Weisstein's World of Mathematics, Multiperfect Number.
Wikipedia, Aliquot sequence.

Cf. A000203, A000396, A005820, A023194, A028982, A028983, A032742, A071189, A371419, A371420, A371421, A371422, A371423.

r[n_] := n/FactorInteger[n][[1, 1]]; a[n_] := r[DivisorSigma[1, n]]; Array[a, 100]

a(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}

a(n) = A032742(A000203(n)).
a(n) = A000203(n)/A071189(n).
a(n) = A000203(n)/2 if n is in A028983 (i.e., n is not in A028982).
a(k) = 1 if and only if k = 1 or k is in A023194.

A248151 Numbers n such that the sum of the divisors of n is not divisible by 4.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 68, 72, 73, 74, 80, 81, 82, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 128, 136, 137, 144, 146, 148, 149, 153, 157, 160, 162, 164, 169, 173, 178, 180

Offset: 1

M. F. Hasler, Oct 02 2014

nonn

The complement of A248150; contains in particular A028982, the squares and twice the squares, for which sigma is odd.

The terms cannot have more than one odd prime factor to an odd power. Therefore this set has asymptotic density zero. The sequence grows faster than .75 n log(n). In particular: a(10) = 17, a(100) = 298, a(10^3) = 4724, a(10^4) = 66533, a(10^5) = 872434, a(10^6) = 10819205. - M. F. Hasler, Apr 26 2017

M. F. Hasler, Table of n, a(n) for n = 1..1000

Cf. A000203 (sum of divisors), A028982, A028983, A248150.

Subsequence of A285801.

Select[Range[200],Mod[DivisorSigma[1,#],4]!=0&] (* Harvey P. Dale, Apr 03 2025 *)

PARI
```
for(n=1,999,sigma(n)%4&&print1(n","))
```

A274397 Positive integers m such that sigma(m) is divisible by 5.

Original entry on oeis.org

8, 19, 24, 27, 29, 38, 40, 54, 56, 57, 58, 59, 72, 76, 79, 87, 88, 89, 95, 104, 108, 109, 114, 116, 118, 120, 128, 133, 135, 136, 139, 145, 149, 152, 158, 168, 171, 174, 177, 178, 179, 184, 189, 190, 199, 200, 203, 209, 216, 218, 228, 229, 232, 236, 237, 239, 247, 248, 261, 264, 266, 267, 269, 270, 278, 280, 285, 290, 295, 296, 297

Offset: 1

M. F. Hasler, Jul 02 2016

nonn

See the subsequence A274685 of odd terms for a remark on frequent pairs of the form (30k-3, 30k-1).
If m is in the sequence and gcd(k,m)=1, then k*m is also in the sequence. One might call "primitive" those terms which are not of this form, i.e., not a "coprime" multiple of an earlier term. The primitive terms are the primes and powers of primes within the sequence, cf. below.
Integers m > 0 where an integer k exists such that A000203(m) = A008587(k). - Felix Fröhlich, Jul 02 2016
For any prime p <> 5 there is an exponent k in {1, 3, 4} (depending on whether p is in A030433, A003631 or A030430) such that p^k is in this sequence. Given these p^k, the sequence consists of all numbers of the form n*p^(q*(k+1)-1) where n is coprime to p and q >= 1. Otherwise said, all numbers m which have some prime factor p with multiplicity q*(k+1)-1, where k = k(p) in {1, 3, 4} as introduced before. - M. F. Hasler, Jul 10 2016

			Some values for a(2^k): We have a(2) = 19, a(4) = 27, a(8) = 54, a(16) = 87, a(32) = 145, a(64) = 270, a(128) = 488, a(256) = 919, a(512) = 1736, a(1024) = 3267, a(2048) = 6258, a(4096) = 12035, a(8192) = 23160, a(16384) = 44878, a(32768) = 87207, a(65536) = 169911, a(131072) = 332009, a(262144) = 650031, a(524288) = 1274569, a(1048576) = 2503510, a(2097152) = 4924370, a(4194304) = 9697475, a(8388608) = 19116191.

Robert Israel, Table of n, a(n) for n = 1..10000
Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 20.
N. J. A. Sloane, Needed: smallest number k with sigma(sigma(k)) = 5k, SeqFan list, Jul 02 2016.

Cf. A000203, A028983 (sigma even), A087943 (sigma = 3k), A248150 (sigma = 4k); A028982 (sigma is odd), A248151 (sigma is not divisible by 4); A272930 (sigma(sigma(k)) = nk).

select(t -> numtheory:-sigma(t) mod 5 = 0, [$1..1000]); # Robert Israel, Jul 12 2016

Select[Range[300], Divisible[DivisorSigma[1, #], 5]&] (* Jean-François Alcover, Apr 09 2019 *)

PARI
```
is(n)=sigma(n)%5==0
```

is(n)=for(i=1,#n=factor(n)~,n[1,i] != 5 && (n[2,i]+1) % [5,4,4,2][n[1,i]%5] == 0 && return(1))

lim_{n->oo} a(k)/k = 2 (conjectured; cf. Examples).

Edited by M. F. Hasler, Jul 10 2016

A334748 Let p be the smallest odd prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller odd primes.

Original entry on oeis.org

3, 6, 5, 12, 15, 10, 21, 24, 27, 30, 33, 20, 39, 42, 7, 48, 51, 54, 57, 60, 35, 66, 69, 40, 75, 78, 45, 84, 87, 14, 93, 96, 55, 102, 105, 108, 111, 114, 65, 120, 123, 70, 129, 132, 135, 138, 141, 80, 147, 150, 85, 156, 159, 90, 165, 168, 95, 174, 177, 28, 183, 186, 189

Offset: 1

Peter Munn, May 09 2020

A permutation of A028983.
A007417 (which has asymptotic density 3/4) lists index n such that a(n) = 3n. The sequence maps the terms of A007417 1:1 onto A145204\{0}, defining a bijection between them.
Similarly, bijections are defined from the odd numbers (A005408) to the nonsquare odd numbers (A088828), from the positive even numbers (A299174) to A088829, from A003159 to the nonsquares in A003159, and from A325424 to the nonsquares in A036668. The latter two bijections are between sets where membership depends on whether a number's squarefree part divides by 2 and/or 3.

			84 = 21*4 has squarefree part 21 (and square part 4). The smallest odd prime absent from 21 = 3*7 is 5 and the product of all smaller odd primes is 3. So a(84) = 84*5/3 = 140.

Permutation of A028983.
Row 3, and therefore column 3, of A331590. Cf. A334747 (row 2).
A007913, A034386, A225546, A284723 are used in formulas defining the sequence.
The formula section details how the sequence maps the terms of A003961, A019565, A070826; and how f(a(n)) relates to f(n) for f = A008833, A048675, A267116; making use of A003986.
Subsequences: A016051, A145204\{0}, A329575.
Bijections are defined that relate to A003159, A005408, A007417, A036668, A088828, A088829, A299174, A325424.

a(n) = {my(c=core(n), m=n); forprime(p=3, , if(c % p, m*=p; break, m/=p)); m;} \\ Michel Marcus, May 22 2020

a(n) = n * p / (A034386(p-1)/2), where p = A284723(A007913(n)).
a(n) = A334747(A334747(n)).
a(n) = A331590(3, n) = A225546(4 * A225546(n)).
a(2*n) = 2 * a(n).
a(A019565(n)) = A019565(n+2).
a(k * m^2) = a(k) * m^2.
a(A003961(n)) = A003961(A334747(n)).
a(A070826(n)) = prime(n+1).
A048675(a(n)) = A048675(n) + 2.
A008833(a(n)) = A008833(n).
A267116(a(n)) = A267116(n) OR 1, where OR denotes the bitwise operation A003986.
a(A007417(n)) = A145204(n+1) = 3 * A007417(n).

A339245 Partrich numbers: positive integers whose square part and squarefree part are divisible by 2 and an odd prime.

Original entry on oeis.org

216, 360, 504, 600, 792, 864, 936, 1000, 1080, 1176, 1224, 1368, 1400, 1440, 1512, 1656, 1944, 1960, 2016, 2088, 2200, 2232, 2376, 2400, 2520, 2600, 2664, 2744, 2808, 2904, 2952, 3000, 3096, 3168, 3240, 3384, 3400, 3456, 3672, 3744, 3800, 3816, 3960, 4000, 4056, 4104, 4200

Offset: 1

Peter Munn, Nov 28 2020

Not named after anyone, partrich numbers have the square part of their odd part, the square part of their even part (A234957), the squarefree part of their odd part and the squarefree part of their even part (A056832) all greater than 1.
Numbers whose odd part and even part are nonsquare and nonsquarefree.
All terms are divisible by 8. If m is present, 2m is absent and 4m is present.
Closed under multiplication by any square and under application of A059896: for n, k >= 1, A059896(a(n), k) is in the sequence.
From Peter Munn, Apr 07 2021: (Start)
The first deficient partrich number is 39304 = 2^3 * 17^3. (ascertained by Amiram Eldar)
The first 7 terms generate Carmichael numbers using the method of Erdős described in A287840.
(End)

			A positive integer is present if and only if it factorizes as 2 times an odd squarefree number > 1, an even square that is a power of 4 and an odd square > 1. This factorization of the initial terms is shown below.
   n  a(n)
   1   216 = 2 *  3 *  4 *  9,
   2   360 = 2 *  5 *  4 *  9,
   3   504 = 2 *  7 *  4 *  9,
   4   600 = 2 *  3 *  4 * 25,
   5   792 = 2 * 11 *  4 *  9,
   6   864 = 2 *  3 * 16 *  9,
   7   936 = 2 * 13 *  4 *  9,
   8  1000 = 2 *  5 *  4 * 25,
   9  1080 = 2 * 15 *  4 *  9,
  10  1176 = 2 *  3 *  4 * 49,
  ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Even Part, Odd Part, Square Part, Squarefree Part.

Subsequence of A008586, A028983, A036554, A036785, A038838, A190892.
Subsequences: A017139, A017643.
Cf. A056832, A059896, A234957, A287840.

q[n_] := Module[{ie = IntegerExponent[n, 2], odd}, ie > 2 && OddQ[ie] && !SquareFreeQ[(odd = n/2^ie)] && !IntegerQ @ Sqrt[odd]]; Select[Range[4200], q] (* Amiram Eldar, Dec 04 2020 *)

A008586 INTERSECT A028983 INTERSECT A036554 INTERSECT A038838.

Asymptotic density is 1/12 - 2/(3 * Pi^2) = 0.01578587757... . (Formula due to Amiram Eldar.)

A049605 Smallest k>1 such that k divides sigma(k*n).

Original entry on oeis.org

6, 3, 2, 6, 2, 2, 2, 3, 6, 2, 2, 2, 2, 2, 2, 6, 2, 3, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 6, 2, 2, 2, 2, 2

Offset: 1

Benoit Cloitre, Jul 26 2002

a(n) = 2, 3 or 6. For any m, a(A028983(m)) = 2. If a(m)=6 then m is a square but if m is a square a(m) is not necessarily 6, first example is 7: a(7^2)=3 (cf. A072864).

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A028983 (locations of 2), A067051 (locations of 3), A072862 (locations of 6).

A049605 := proc(n)
    for k from 2 do
        if modp(numtheory[sigma](k*n),k) = 0 then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Oct 26 2015

sk[n_]:=Module[{k=2},While[!Divisible[DivisorSigma[1,k*n],k],k++];k]; sk /@ Range[110] (* Harvey P. Dale, Jan 04 2015 *)

a(n) = {k = 2; while(sigma(k*n) % k, k++); k ;} \\ Michel Marcus, Nov 21 2013

A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

Original entry on oeis.org

9, 18, 25, 36, 49, 50, 72, 81, 98, 100, 121, 144, 162, 169, 196, 200, 225, 242, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1058, 1089, 1152, 1156, 1225, 1250, 1296, 1352, 1369

Offset: 1

Benoit Cloitre, Apr 18 2002

Previous name: sum(d|n,6d/(2+mu(d))) is odd, where mu(.) is the Moebius function, A008683.
From Peter Munn, Jul 06 2020: (Start)
Numbers that have an odd number of odd nonsquarefree divisors.
[Proof of equivalence to the name, where m denotes a positive integer:
(1) These properties are equivalent: (a) m has an even number of odd squarefree divisors; (b) m has a nontrivial odd part.
(2) These properties are equivalent: (a) m has an odd number of odd divisors; (b) the odd part of m is square.
(3) m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are both true or both false.
(4) The trivial odd part, 1, is a square, so (1)(b) and (2)(b) cannot both be false, which (from (1), (2)) means (1)(a) and (2)(a) cannot both be false.
(5) From (3), (4), m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are true.
(6) m satisfies the condition in the name if and only if (1)(b) and (2)(b) are true, which (from (1), (2)) is equivalent to (1)(a) and (2)(a) being true, and hence from (5), to m satisfying the condition at the start of this comment.]
(End)
Numbers whose sum of non-unitary divisors (A048146) is odd. - Amiram Eldar, Sep 16 2024

			To determine the odd part of 18, remove all factors of 2, leaving 9. 9 is a nontrivial square, so 18 is in the sequence. - _Peter Munn_, Jul 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Odd part.

A000265, A008683 are used in definitions of this sequence.
Lists of numbers whose odd part satisfies other conditions: A028982 (square), A028983 (nonsquare), A029747 (less than 6), A029750 (less than 8), A036349 (even number of prime factors), A038550 (prime), A070776 U {1} (power of a prime), A072502 (square of a prime), A091067 (has form 4k+3), A091072 (has form 4k+1), A093641 (noncomposite), A105441 (composite), A116451 (greater than 4), A116882 (less than or equal to even part), A116883 (greater than or equal to even part), A122132 (squarefree), A229829 (7-rough), A236206 (11-rough), A260488\{0} (has form 6k+1), A325359 (proper prime power), A335657 (odd number of prime factors), A336101 (prime power).
Cf. A048146, A091476.

Select[Range[1000], (odd = #/2^IntegerExponent[#, 2]) > 1 && IntegerQ @ Sqrt[odd] &] (* Amiram Eldar, Sep 29 2020 *)

upto(n) = { my(res = List()); forstep(i = 3, sqrtint(n), 2, for(j = 0, logint(n\i^2, 2), listput(res, i^2<David A. Corneth, Sep 28 2020

Sum_{n>=1} 1/a(n) = 2 * Sum_{k>=1} 1/(2*k+1)^2 = Pi^2/4 - 2 = A091476 - 2 = 0.467401... - Amiram Eldar, Feb 18 2021

New name from Peter Munn, Jul 06 2020

A183300 Positive integers not of the form 2n^2.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

Clark Kimberling, Jan 03 2011

Complement of A001105.

Integers whose number of even divisors (A183063) is even (for a proof, see A001105, the complement of this sequence), hence odd numbers (A005408) are a subsequence. - Bernard Schott, Sep 15 2021

			10 is in the sequence since 2*2^2=8 < 10 < 2*3^2=18.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A005408, A183063.

Cf. A001105 (number of even divisors is odd), A028982 (number of odd divisors is odd), A028983 (number of odd divisors is even), this sequence (number of even divisors is even).

[n: n in [0..100] | not IsSquare(n/2)]; // Bruno Berselli, Dec 17 2013

A183300:=n->if type(sqrt(2*n)/2, integer) then NULL; else n; fi; seq(A183300(n), n=1..100); # Wesley Ivan Hurt, Dec 17 2013

a = 2; b = 0;
F[n_] := a*n^2 + b*n;
R[n_] := (n/a + ((b - 1)/(2a))^2)^(1/2);
G[n_] := n - 1 + Ceiling[R[n] - (b - 1)/(2a)];
Table[F[n], {n, 60}]
Table[G[n], {n, 100}] (* Clark Kimberling *)
r[n_] := Reduce[n == 2*k^2, k, Integers]; Select[Range[100], r[#] === False &] (* Jean-François Alcover, Dec 17 2013 *)
max = 100; Complement[Range[max], 2 Range[Ceiling[Sqrt[max/2]]]^2] (* Alonso del Arte, Dec 17 2013 *)
Module[{nn=10,f},Complement[Range[2nn^2],2Range[nn]^2]] (* Harvey P. Dale, Sep 06 2023 *)

is(n)=!issquare(n/2) \\ Charles R Greathouse IV, Sep 02 2015

a(n)=my(k=sqrtint(n\2)+n); if(k-sqrtint(k\2)Charles R Greathouse IV, Sep 02 2015

from math import isqrt
def A183300(n): return n+(m:=isqrt(n>>1))+(n>(m+1)*((m<<1)+1)) # Chai Wah Wu, Aug 04 2025

a(n) = n + floor(sqrt(n/2) + 1/4). - Ridouane Oudra, Jan 26 2023

a(n) = n+m+1 if n>(m+1)*(2m+1) and a(n) = n+m otherwise where m = floor(sqrt(n/2)). - Chai Wah Wu, Aug 04 2025

Name clarified by Wesley Ivan Hurt, Dec 17 2013

A204827 Deficient numbers with even sum of divisors.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 99, 101, 103

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

Numbers m from A005100 such that sigma(m) = A000203(m) is even.

Complement of A204826 with respect to A005100 (deficient numbers).

			Deficient number 15 is in sequence because sigma(15) = 24 (even number).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Deficient Number

Cf. A000203, A005100, A028983.

Cf. A156903 (abundant numbers with odd sum of divisors), A204825 (abundant numbers with even sum of divisors), A204826 (deficient numbers with odd sum of divisors).

dnesQ[n_]:=Module[{s=DivisorSigma[1,n]},s<2n&&EvenQ[s]]; Select[ Range[ 120],dnesQ] (* Harvey P. Dale, Nov 23 2014 *)

cp's OEIS Frontend

A152678 Even members of A000203.

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A371418 The largest aliquot divisor of the sum of divisors of n; a(1) = 1.

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A248151 Numbers n such that the sum of the divisors of n is not divisible by 4.

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A274397 Positive integers m such that sigma(m) is divisible by 5.

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A334748 Let p be the smallest odd prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller odd primes.

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A339245 Partrich numbers: positive integers whose square part and squarefree part are divisible by 2 and an odd prime.

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A049605 Smallest k>1 such that k divides sigma(k*n).

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