A066191 -id:A066191 - cp's OEIS frontend

A091570 Sum of odd proper divisors of n. Sum of the odd divisors of n that are less than n.

0, 1, 1, 1, 1, 4, 1, 1, 4, 6, 1, 4, 1, 8, 9, 1, 1, 13, 1, 6, 11, 12, 1, 4, 6, 14, 13, 8, 1, 24, 1, 1, 15, 18, 13, 13, 1, 20, 17, 6, 1, 32, 1, 12, 33, 24, 1, 4, 8, 31, 21, 14, 1, 40, 17, 8, 23, 30, 1, 24, 1, 32, 41, 1, 19, 48, 1, 18, 27, 48, 1, 13, 1, 38, 49

Offset: 1

Mohammad K. Azarian, Mar 04 2004

			The sum of odd divisors of 9 that are less than 9 is 1 + 3 = 4.

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

Cf. A000035, A000593, A013661, A066191.

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), this sequence (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

f[2, e_] := 1; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n, 0]; Array[a, 75] (* Amiram Eldar, Oct 11 2023 *)

a(n) = sumdiv(n , d, (d%2) * (dMichel Marcus, Jan 14 2014

If n is odd, a(n) = A000593(n) - n; if n is even, a(n) = A000593(n). - Michel Marcus, Jan 14 2014
G.f.: Sum_{k>=1} (2*k-1) * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Apr 13 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)-1)/4 = 0.1612335167... . - Amiram Eldar, Oct 11 2023

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

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));

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A091570 Sum of odd proper divisors of n. Sum of the odd divisors of n that are less than n.

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

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