A113285 -id:A113285 - cp's OEIS frontend

A114529 Let S(n)=sigma(|n|)/2-n; sequence gives numbers n such that S(S(S(S(n))))=n. May be called {1,2}-sociable number of orders 1 or 2 or 4.

41, 929, 1301, 30240, 32760, 260609, 1441440, 1860768, 2178540, 3205440, 3378240, 3423420, 3914820, 4029480, 4437720, 5738040, 6093360, 6807240, 7136640, 7239120, 7551360, 9402120, 10204740, 12270720, 12405120, 12942720, 13495680, 14627340, 14725620

Offset: 1

Yasutoshi Kohmoto, Feb 15 2006

nonn

Eric Weisstein's World of Mathematics, Sociable Numbers

Cf. A113285, A114528.

a(7)-a(29) from Charles R Greathouse IV, Nov 12 2010.

A075701 a(1)=1, a(n+1)=sigma(a(n))-2*a(n).

Original entry on oeis.org

1, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6

Offset: 1

Benoit Cloitre, Oct 02 2002

sign

Taking any nonperfect number as initial value, does the map x->sigma(x)-2x lead to the cycle (-1,3,-2,7,-6,24,12,4) if during the iteration no perfect number is reached? Example: 124 -> -24 -> 108 -> 64 -> -1 -> 3 -> -2 -> 7 -> -6 -> 24 -> 12 -> 4 and the cycle (-1,3,-2,7,-6,24,12,4) is reached.

There appear to be lots of other cycles, for example the numbers in A005820 are cycles of length one. For longer cycles refer to the discussion in links. - Hans Havermann, Jul 21 2013

Peter Luschny, Open questions posed in OEIS #1
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A146556, A005820, A069085, A113285.

NestList[DivisorSigma[1, #]-2#&, 1, 94]  (* Peter Luschny, Jul 17 2013 *)
Join[{1},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{-1, 3, -2, 7, -6, 24, 12, 4},93]] (* Ray Chandler, Aug 25 2015 *)

Periodic with period (-1, 3, -2, 7, -6, 24, 12, 4) of length 8.

A114528 Let S(n)=sigma(|n|)-3*n; sequence gives numbers n such that S(S(S(S(n))))=n. May be called {3,1}-Sociable number of orders 1 or 2 or 4.

Original entry on oeis.org

1248, 1596, 28272, 30240, 32760, 463296, 678032, 1906128, 2178540, 4694328, 4697616, 4698072, 11110976, 12865770, 23569920, 30998250, 31235904, 37501072, 45532800, 63723600, 76980288, 95801088, 142990848, 146078592, 163032720, 614533696, 1044244800, 1379454720

Offset: 1

Yasutoshi Kohmoto, Feb 15 2006

nonn

Eric Weisstein's World of Mathematics, Sociable Numbers

Cf. A113285, A114529. Contains A069146 as a subsequence.

S(n)=sigma(abs(n))-3*n
is(n)=S(S(S(S(n))))==n \\ Charles R Greathouse IV, Jan 16 2012

a(5)-a(12) from Charles R Greathouse IV, Jan 20 2010

a(13)-a(28) from Donovan Johnson, Jan 16 2012

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.

cp's OEIS Frontend

A114529 Let S(n)=sigma(|n|)/2-n; sequence gives numbers n such that S(S(S(S(n))))=n. May be called {1,2}-sociable number of orders 1 or 2 or 4.

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A075701 a(1)=1, a(n+1)=sigma(a(n))-2*a(n).

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A114528 Let S(n)=sigma(|n|)-3*n; sequence gives numbers n such that S(S(S(S(n))))=n. May be called {3,1}-Sociable number of orders 1 or 2 or 4.

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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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