A091570 -id:A091570 - cp's OEIS frontend

A066191 Numbers k such that the sum of the odd aliquot parts of k divides k.

2, 3, 4, 5, 7, 8, 11, 12, 13, 16, 17, 19, 23, 24, 29, 31, 32, 37, 41, 43, 47, 48, 53, 56, 59, 61, 64, 67, 71, 73, 79, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 120, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 192, 193, 197, 199, 211, 223

Offset: 1

Robert G. Wilson v, Dec 15 2001

			12 is in the sequence because the odd aliquot parts of 12 are {1,3} and their sum divides 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Disjoint union of A000040 and A066192.

Cf. A091570.

with(numtheory):soa:=proc(n) local div,s,j: div:=convert(divisors(n),list): s:=0: for j from 1 to nops(div)-1 do if div[j] mod 2=1 then s:=s+div[j] else s:=s: fi: od: end: p:=proc(n) if type(n/soa(n),integer)=true then n else fi end: seq(p(n),n=1..240); # Emeric Deutsch, Feb 26 2005

Do[ d = Drop[ Divisors[ n ], -1 ]; l = Length[ d ]; od = 1; k = 2; While[ k <= l, If[ OddQ[ d[ [ k ] ] ], od = od + d[ [ k ] ] ]; k++ ]; If[ IntegerQ[ n/od ], Print[ n ] ], {n, 2, 200} ]
Select[Range[2, 225], Divisible[#, DivisorSigma[1, #/2^IntegerExponent[#, 2]] - If[OddQ[#], #, 0]] &] (* Amiram Eldar, Apr 27 2025 *)

{ n=0; for (m=2, 10^9, d=divisors(m); s=1; for (i=2, numdiv(m) - 1, if (d[i]%2, s += d[i])); if (s > 0 && m%s == 0, write("b066191.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 05 2010

isok(n) = !(n % sumdiv(n, d, d*(d%2)*(d!=n))); \\ Michel Marcus, Apr 06 2015

isok(k) = if(k == 1, 0, !(k % (sigma(k >> valuation(k, 2)) - if(k%2, k)))); \\ Amiram Eldar, Apr 27 2025

More terms from Emeric Deutsch, Feb 26 2005

A066192 Composite numbers k such that the sum of the odd aliquot parts of k divides k.

Original entry on oeis.org

4, 8, 12, 16, 24, 32, 48, 56, 64, 96, 112, 120, 128, 192, 224, 240, 256, 384, 448, 480, 512, 528, 672, 768, 896, 960, 992, 1024, 1056, 1344, 1456, 1536, 1792, 1920, 1984, 2048, 2112, 2160, 2208, 2688, 2912, 3072, 3584, 3840, 3968, 4096, 4224, 4320, 4416

Offset: 1

Robert G. Wilson v, Dec 15 2001

nonn

From Amiram Eldar, Apr 27 2025: (Start)
If there is an odd term in this sequence it must be an odd perfect number (A000396). If k is an odd term then d = sigma(k)-k divides k. If d < k then sigma(k) = k + d with d being an aliquot divisor of k which is possible only if k is prime and d = 1. Therefore, d = k and k must be an odd perfect number.
This sequence is infinite because if k is a term then 2*k is also a term. The primitive terms are in A383428. (End)

Amiram Eldar, Table of n, a(n) for n = 1..4905 (terms below 10^11; terms 1..1000 from Harry J. Smith)
Index entries for sequences where any odd perfect numbers must occur.

Intersection of A002808 and A066191.

Cf. A000396, A091570, A383428.

Do[ d = Drop[ Divisors[ n ], -1 ]; l = Length[ d ]; od = 1; k = 2; While[ k <= l, If[ OddQ[ d[ [ k ] ] ], od = od + d[ [ k ] ] ]; k++ ]; If[ !PrimeQ[ n ] && IntegerQ[ n/od ], Print[ n ] ], {n, 2, 10^4} ]
Select[Range[4500], CompositeQ[#] && Divisible[#, DivisorSigma[1, #/2^IntegerExponent[#, 2]] - If[OddQ[#], #, 0]] &] (* Amiram Eldar, Apr 27 2025 *)

{ n=0; for (m=4, 10^9, if (isprime(m), next); d=divisors(m); s=1; for (i=2, numdiv(m) - 1, if (d[i]%2, s += d[i])); if (s > 0 && m%s == 0, write("b066192.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 05 2010

isok(k) = if(k == 1, 0, !isprime(k) && !(k % (sigma(k >> valuation(k, 2)) - if(k%2, k)))); \\ Amiram Eldar, Apr 27 2025

A091818 Sum of the even proper divisors of 2n. Sum of the even divisors of 2n that are less than 2n.

Original entry on oeis.org

0, 2, 2, 6, 2, 12, 2, 14, 8, 16, 2, 32, 2, 20, 18, 30, 2, 42, 2, 44, 22, 28, 2, 72, 12, 32, 26, 56, 2, 84, 2, 62, 30, 40, 26, 110, 2, 44, 34, 100, 2, 108, 2, 80, 66, 52, 2, 152, 16, 86, 42, 92, 2, 132, 34, 128, 46, 64, 2, 216, 2, 68, 82, 126, 38, 156, 2, 116

Offset: 1

Mohammad K. Azarian, Mar 07 2004

			The sum of the even divisors of 18 that are less than 18 is 8 = 2+6.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001065, A032741, A091570, A074400, A173455.

Table[Total[Select[Most[Divisors[2 n]],EvenQ]],{n,70}] (* Harvey P. Dale, Apr 28 2023 *)

a(n) = sumdiv(2*n, d, !(d%2) * d * (d<2*n)); \\ Michel Marcus, Jan 14 2014

from sympy import divisors
def a(n): return sum(d for d in divisors(2*n) if d%2==0) - 2*n
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Oct 30 2017

a(n) = A074400(2n) - 2n. - Michel Marcus, Jan 14 2014
a(n) = Sum_{d|2n, d<2n, d even} d. - Wesley Ivan Hurt, Mar 02 2022
a(n) = 2 * A001065(n). - Alois P. Heinz, Mar 02 2022

More terms from Michel Marcus, Jan 14 2014

A293901 Sum of proper divisors of n of the form 4k+1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 10, 1, 6, 1, 1, 1, 1, 6, 14, 10, 1, 1, 6, 1, 1, 1, 18, 6, 10, 1, 1, 14, 6, 1, 22, 1, 1, 15, 1, 1, 1, 1, 31, 18, 14, 1, 10, 6, 1, 1, 30, 1, 6, 1, 1, 31, 1, 19, 34, 1, 18, 1, 6, 1, 10, 1, 38, 31, 1, 1, 14, 1, 6, 10, 42, 1, 22, 23, 1, 30, 1, 1, 60, 14, 1, 1, 1, 6, 1, 1, 50, 43, 31, 1, 18, 1, 14, 27

Offset: 1

Antti Karttunen, Oct 19 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A050449, A091570, A293451, A293903.

a[n_] := DivisorSum[n, # &, # < n && Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Nov 27 2023 *)

PARI
```
A293901(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, d

a(n) = A091570(n) - A293903(n).

G.f.: Sum_{k>=1} (4*k-3) * x^(8*k-6) / (1 - x^(4*k-3)). - Ilya Gutkovskiy, Apr 14 2021

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/48 - 1/8 = 0.0806167... . - Amiram Eldar, Nov 27 2023

A293903 Sum of proper divisors of n of the form 4k+3.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 7, 3, 0, 0, 3, 0, 0, 10, 11, 0, 3, 0, 0, 3, 7, 0, 18, 0, 0, 14, 0, 7, 3, 0, 19, 3, 0, 0, 10, 0, 11, 18, 23, 0, 3, 7, 0, 3, 0, 0, 30, 11, 7, 22, 0, 0, 18, 0, 31, 10, 0, 0, 14, 0, 0, 26, 42, 0, 3, 0, 0, 18, 19, 18, 42, 0, 0, 30, 0, 0, 10, 0, 43, 3, 11, 0, 18, 7, 23, 34, 47, 19, 3, 0, 7, 14, 0, 0, 54, 0, 0, 60

Offset: 1

Antti Karttunen, Oct 19 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A050452, A091570, A293513, A293901.

Array[DivisorSum[#, # &, Mod[#, 4] == 3 &] - Boole[Mod[#, 4] == 3] # &, 105] (* Michael De Vlieger, Oct 23 2017 *)

PARI
```
A293903(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, d

a(n) = A091570(n) - A293901(n).

G.f.: Sum_{k>=1} (4*k-1) * x^(8*k-2) / (1 - x^(4*k-1)). - Ilya Gutkovskiy, Apr 14 2021

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/48 - 1/8 = 0.0806167... . - Amiram Eldar, Nov 27 2023

A383428 Primitive terms in A066192: number k such that k is a term of A066192 and k/2 is not.

Original entry on oeis.org

4, 12, 56, 120, 528, 672, 992, 1456, 2160, 2208, 6720, 9024, 9120, 11904, 13104, 16256, 17472, 24800, 29568, 55104, 55552, 73440, 90816, 95040, 119040, 120960, 121024, 123648, 131040, 146688, 151680, 174720, 195072, 223104, 297600, 397440, 399616, 445536, 505344

Offset: 1

Amiram Eldar, Apr 27 2025

nonn

If a(1) = 1 instead of 4, then this will be the sequence of primitive terms in A069519.
If k is a term then 2^m * k is a term in A066192 for all m >= 0.
If there is an odd term in this sequence it must be an odd perfect number (A000396). See the comments in A066192.
Except for 4, numbers k such that A091570(k) | k and k/A091570(k) is odd.

Amiram Eldar, Table of n, a(n) for n = 1..755 (terms below 10^11)
Index entries for sequences where any odd perfect numbers must occur.

Subsequence of A066191 and A066192.

Cf. A000396, A069519, A091570.

q[n_] := Module[{s = DivisorSigma[1, n/2^IntegerExponent[n, 2]] - If[OddQ[n], n, 0]}, Divisible[n, s] && OddQ[n/s]]; Select[Range[550000], # == 4 || (CompositeQ[#] && q[#]) &]

isok(k) = if(k == 1 || isprime(k), 0, if(k == 4, 1, my(s = sigma(k >> valuation(k, 2)) - if(k%2, k)); !(k % s) && (k/s) % 2));

A385349 Product of odd proper divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 3, 1, 7, 15, 1, 1, 27, 1, 5, 21, 11, 1, 3, 5, 13, 27, 7, 1, 225, 1, 1, 33, 17, 35, 27, 1, 19, 39, 5, 1, 441, 1, 11, 2025, 23, 1, 3, 7, 125, 51, 13, 1, 729, 55, 7, 57, 29, 1, 225, 1, 31, 3969, 1, 65, 1089, 1, 17, 69, 1225, 1, 27, 1, 37, 5625

Offset: 1

Ilya Gutkovskiy, Jun 26 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product.

Cf. A007955, A007956, A091570 (similar for sum), A136655, A385350 (fixed points).

a:= n-> mul(`if`(d::odd, d, 1), d=numtheory[divisors](n) minus {n}):
seq(a(n), n=1..75);  # Alois P. Heinz, Jun 27 2025

a[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Table[a[n], {n, 75}]

a(n) = my(m = n >> valuation(n,2), d = numdiv(m)); if(d % 2, sqrtint(m)^d, m^(d/2)) / if(m < n, 1, n); \\ Amiram Eldar, Jun 27 2025

from math import isqrt
from sympy import divisor_count
def A385349(n):
    d = divisor_count(m:=n>>(~n&n-1).bit_length())
    k = isqrt(m)**d if d&1 else m**(d>>1)
    return k//n if n&1 else k # Chai Wah Wu, Jun 27 2025

a(n) = Product_{d|n, d < n, d odd} d.

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cp's OEIS Frontend

A066191 Numbers k such that the sum of the odd aliquot parts of k divides k.

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A066192 Composite numbers k such that the sum of the odd aliquot parts of k divides k.

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A091818 Sum of the even proper divisors of 2n. Sum of the even divisors of 2n that are less than 2n.

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A293901 Sum of proper divisors of n of the form 4k+1.

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A293903 Sum of proper divisors of n of the form 4k+3.

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A383428 Primitive terms in A066192: number k such that k is a term of A066192 and k/2 is not.

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A385349 Product of odd proper divisors of n.

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