A371920 -id:A371920 - cp's OEIS frontend

A371952 Weird numbers whose abundance is also a weird number.

12110, 1596770, 1629364, 4265590, 4442690, 6015310, 7122290, 7142030, 8438990, 8837570, 8877890, 9667910, 9989770, 10106810, 10867570, 11130910, 11306470, 11511290, 12008710, 12096490, 12623170, 12915770, 13281170, 13390510, 13852090, 14144690, 14583590, 14817670

Offset: 1

Amiram Eldar, Apr 14 2024

nonn

Terms k of A006037 such that A033880(k) is also a term of A006037.

			12110 is a term because it is a weird number, and A033880(12110) = sigma(12110) - 2*12110 = 836 is also a weird number.

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

Subsequence of A006037 and A371920.
A371953 is a subsequence.
Cf. A000203 (sigma), A033880 (abundance), A371950.

With[{weirds = Import["https://oeis.org/a006037/b006037.txt", "Table"][[;; , 2]]}, Select[weirds, (ab = DivisorSigma[1, #] - 2*#) <= Last[weirds] && MemberQ[weirds, ab] &]]

A371921 The number of iterations of the map x -> A033880(x) starting at n until the a nonpositive number is reached, or 0 if this does not happen.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Apr 12 2024

Analogous to A098007 with A033880(n) = sigma(n) - 2*n instead of A001065(n) = sigma(n) - n.

			a(n) = 0 if the iterations that start at n are entering a cycle. Examples of cycles are:
  1) Cycles of length 1: the triperfect numbers (A005820), 120, 672, 523776, ..., which are the fixed points of A033880. The triperfect numbers can be reached from other values of n, e.g., 276, 448, 486, 510, 702, ... .
  2) Cycles of length 2: the only known cycle is (45840, 51168) (see A069085). It can be reached from other values of n, e.g., 32130, 39420, 45480, 66300, ... .
  3) Cycles of length 3: the least cycle is (243732672, 271303776, 256786848). It is first reached from n = 107689320.
  4) Cycles of length 4: the least cycle is (65071776, 82842816, 89761152, 77260656). It can be reached from other values of n, e.g., 33623940, 41132280, 42825888, ... . The next cycle of length 4 is (985948800, 1381340160, 2183133696, 1489384608).

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

Cf. A000203 (sigma), A001065, A033880, A098007.

Cf. A005101, A005820, A069085, A113285, A263837, A371920.

ab[n_] := Module[{k}, If[n < 1, 0, k = DivisorSigma[1, n] - 2*n; If[k < 1, 0, k]]]; a[n_] := Module[{s = NestWhileList[ab, n, UnsameQ, All]}, If[s[[-1]] == 0, Length[s] - 2, 0]]; Array[a, 120]

ab(n) = {my(k); if(n < 1, 0, k = sigma(n) - 2*n; if(k < 1, 0, k));}
a(n) = {my(t = 0); until(bittest(t, n = ab(n)), t += 1<M. F. Hasler at A098007

a(n) = 1 if and only if n is nonabundant (A263837).
If a(n) > 0 then:
  a(n) > 1 if n is abundant (A005101).
  a(n) > 2 if n is in A371920.

A371953 Weird numbers whose abundance is a weird number with weird abundance (A371952).

Original entry on oeis.org

13516958630, 13662237610, 14483820470, 16161187910, 16266406870, 17187072770, 17779204310, 18055129190, 19265719130, 20027984690, 20291032090, 20945054548, 21036318170, 21395831170, 21983936030, 22134086030, 22335605390, 23837897510, 24709171970, 25630894310, 26286582910

Offset: 1

Amiram Eldar, Apr 14 2024

nonn

Terms k of A006037 such that A033880(k) and A033880(A033880(k)) are also terms of A006037.

Are there numbers k such that k, A033880(k), A033880(A033880(k)), and A033880(A033880(A033880(k))) are all terms of A006037? There are none below 10^11.

			13516958630 is a term because it is a weird number, A033880(13516958630) = sigma(13516958630) - 2*13516958630 = 833308916 is a weird number, and A033880(833308916) = sigma(833308916) - 2*833308916 = 7975928 is also a weird number.

Amiram Eldar, Table of n, a(n) for n = 1..146 (terms below 10^11)
Wikipedia, Weird number.

Subsequence of A006037, A371920 and A371952.

Cf. A000203 (sigma), A033880 (abundance).

cp's OEIS Frontend

A371952 Weird numbers whose abundance is also a weird number.

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A371921 The number of iterations of the map x -> A033880(x) starting at n until the a nonpositive number is reached, or 0 if this does not happen.

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A371953 Weird numbers whose abundance is a weird number with weird abundance (A371952).

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