A097024 -id:A097024 - cp's OEIS frontend

A097025 Numbers n such that both sigma(n) and sigma(sigma(n)) are odd numbers.

1, 81, 400, 32400, 195938, 224450, 1705636, 3648100, 13645088, 15870978, 18180450, 29184800, 138156516, 295496100, 1055340196, 1105252128, 1476326929, 2263475776, 2323432804, 2363968800, 2592846400, 2661528100, 2686005218, 2917410498, 3564550178

Offset: 1

Labos Elemer, Aug 24 2004

nonn

Only members of A028982 are candidates. - Robert G. Wilson v, Aug 27 2004

Question: iterating sigma()=A000203, how many iterates can be odd numbers?

			n = 1910^2, sigma(n) = 2821^2, sigma(sigma(n)) = 10357983.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A008847, A000203, A097024, A028982.

t = Sort[ Flatten[ Table[{n^2, 2n^2}, {n, 36650}]]]; a = {}; Do[ If[ OddQ[ DivisorSigma[1, DivisorSigma[1, t[[n]] ]]], AppendTo[a, t[[n]] ]], {n, 2*10^6}]; a (* Robert G. Wilson v, Aug 27 2004 *)

Edited and extended by Robert G. Wilson v, Aug 27 2004

A098187 Initial seeds x which will enter a cycle of length 4 under the iteration of x -> A063919(x), the sum of proper unitary divisors.

Original entry on oeis.org

81570, 114270, 137046, 169998, 177906, 182082, 182094, 185190, 194574, 194586, 201642, 203442, 204420, 204540, 212466, 212874, 213870, 219306, 219318, 230874, 231438, 231834, 231846, 232626, 237678, 238134, 242634, 258882, 259338, 259350

Offset: 1

Labos Elemer, Sep 02 2004

nonn

The sequence is the attractor-basin of set of {C4} cycles belonging to this iteration.

The {C4} attractor-set is displayed separately in A098188.

			81570 is in the sequence because its track under the iterated map is 81570, 114270, 182082, 182094, 232626, 237678, 305682, 352878, 360978, 403662, 420738, [420750, 395730, 395910, 420570], 420750.., where the cycle is indicated by brackets. The 4 recurrent terms appear after 11 transients for this case.

Cf. A098185, A098186, A002827, A097024, A097030.

A369895 Irregular triangle of iteration steps of A063919 until the end of the terminal cycle is reached, read by rows.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 7, 1, 8, 1, 9, 1, 10, 8, 1, 11, 1, 12, 8, 1, 13, 1, 14, 10, 8, 1, 15, 9, 1, 16, 1, 17, 1, 18, 12, 8, 1, 19, 1, 20, 10, 8, 1, 21, 11, 1, 22, 14, 10, 8, 1, 23, 1, 24, 12, 8, 1, 25, 1, 26, 16, 1, 27, 1, 28, 12, 8, 1, 29, 1, 30, 42, 54

Offset: 1

Hartmut F. W. Hoft, Feb 04 2024

			The beginning of the irregular triangle showing 3 terminal cycles ( 1 ), ( 6 ) and ( 30 42 54 ):
  1
  2    1
  3    1
  4    1
  5    1
  6
  7    1
  ...
  14  10   8   1
  ...
  30  42  54
  31  1
  ...
Row 1230 contains a non-monotone iteration that ends in the 5-cycle starting at A097035(3):
1230, 1794, 2238, 2250, 1530, 1710, {1890, 2142, 2178, 1482, 1878 }.

Cf. A002827, A063919, A097024, A097030, A097031, A097034, A097035, A318882, A318883.

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#] == 1 &]] - n /; n > 1
iter[k_] := Most[NestWhileList[a063919, k, UnsameQ, All]]
a369895[n_] := Map[iter, Range[n]]
a369895[30] (* irregular triangle *)
Flatten[a369895[30]] (* sequence data *)

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A097025 Numbers n such that both sigma(n) and sigma(sigma(n)) are odd numbers.

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A098187 Initial seeds x which will enter a cycle of length 4 under the iteration of x -> A063919(x), the sum of proper unitary divisors.

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A369895 Irregular triangle of iteration steps of A063919 until the end of the terminal cycle is reached, read by rows.

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Mathematica