A274790 -id:A274790 - cp's OEIS frontend

A362401 Numbers in the range of A162296, where A162296(n) is the sum of divisors of n that have a square factor larger than 1.

0, 4, 9, 12, 16, 24, 25, 27, 28, 32, 36, 48, 49, 54, 56, 60, 72, 75, 79, 80, 96, 100, 108, 112, 117, 120, 121, 124, 126, 128, 144, 147, 150, 152, 162, 168, 169, 176, 180, 183, 192, 196, 199, 200, 216, 224, 240, 248, 252, 268, 270, 272, 288, 289, 294, 296, 300

Offset: 1

Amiram Eldar, Apr 18 2023

nonn

Possible values of A162296 in increasing order.

			0 is a term since A162296(k) = 0 if k is squarefree (A005117).

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

Cf. A005117, A162296.

Similar sequences: A078923, A002191, A002202, A002174, A274790.

s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; m = 300; Select[Union[Array[s, m]], # <= m &]

s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = select(x -> (x < kmax), Set(vector(kmax, k, s(k))))

A065768 Numbers that are sums of divisors of the odd squares; Intersection of A065764 and A065766, written in ascending order and duplicates removed.

Original entry on oeis.org

1, 13, 31, 57, 121, 133, 183, 307, 381, 403, 553, 741, 781, 871, 993, 1093, 1407, 1723, 1729, 1767, 1893, 2257, 2379, 2801, 2863, 3541, 3751, 3783, 3991, 4123, 4557, 4953, 5113, 5403, 5673, 6321, 6897, 6973, 7189, 7581, 8011, 9507, 9517, 9841, 10153

Offset: 1

Labos Elemer, Nov 19 2001

nonn

Terms are the sum of the odd divisors (A000593) of the odd squares (A016754), written in ascending order. Subsequence of the odd terms of A274790. - Timothy L. Tiffin, Feb 12 2022

Equally, the sum of divisors (A000203) as only odd divisors are present in odd squares. - Antti Karttunen, Dec 22 2024

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

Cf. A000203, A000593, A016754, A065764, A065766, A274790.

Sequence A379223 sorted into ascending order, with duplicates removed.

f1[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); f2[2, e_] := (4^(e + 1) - 1)/3; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; seq[max_] := Intersection[Select[Array[s1, max], # < max^2 &], Select[Array[s2, max], # < max^2 &]]; seq[101] (* Amiram Eldar, Aug 24 2024 *)

Old definition clarified and Timothy L. Tiffin's comment adopted as a new primary definition - Antti Karttunen, Dec 22 2024

A274793 Numbers not in the range of the sum of odd divisors function.

Original entry on oeis.org

2, 3, 5, 7, 9, 10, 11, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 39, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100

Offset: 1

Timothy L. Tiffin, Jul 07 2016

nonn

Numbers which do not appear in A000593; that is, there is no positive integer N whose sum of odd divisors is equal to a(n) for any n.

The sum of odd divisors of x is equal to sigma(x) if and only if x is odd. So, there are numbers in the range of sigma(x) that are not in the range of the sum of odd divisors function.

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

Cf. A000593, supersequence of A007369, A274790 (complement).

TakeWhile[Complement[Range@ #, Union@ Table[Total@ Select[Divisors@ n, OddQ], {n, 2 #}]], Function[k, k <= #]] &@ 100 (* Michael De Vlieger, Jul 07 2016 *)

cp's OEIS Frontend

A362401 Numbers in the range of A162296, where A162296(n) is the sum of divisors of n that have a square factor larger than 1.

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A065768 Numbers that are sums of divisors of the odd squares; Intersection of A065764 and A065766, written in ascending order and duplicates removed.

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A274793 Numbers not in the range of the sum of odd divisors function.

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