A166227 Let dsf(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} is the list of the digits of an integer n. dsf(793312220) = 388244100 and dsf(388244100) = 33554978, ..., dsf(387467199) = 793312220, ... in this way these 40 numbers make a loop for the function dsf.

793312220, 388244100, 33554978, 405027808, 34381363, 16824237, 17647707, 3341086, 16824184, 33601606, 140025, 3388, 33554486, 16830688, 50424989, 791621836, 405114593, 387427281, 35201810, 16780376, 18517643, 17650825, 17653671, 1743552, 830081, 33554462, 53476, 873607, 18470986, 421845378, 34381644, 16824695, 404294403, 387421546, 17651084, 17650799, 776537847, 20121452, 3396, 387467199, 793312220

Offset

1,1

Comments

In fact there are only 8 loops among all the nonnegative integers for the "dsf" function that we defined. We have discovered this fact through calculations using Mathematica and general-purpose languages.

Links

Table of n, a(n) for n=1..41.

Ryohei Miyadera, Curious Properties of an Iterative Process, Mathsource, Wolfram Library Archive.

Shoei Takahashi, Unchone Lee, Hikaru Manabe, Aoi Murakami, Daisuke Minematsu, Kou Omori, and Ryohei Miyadera, Curious Properties of Iterative Sequences, arXiv:2308.06691 [math.GM], 2023.

Formula

Let dsf(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} is the list of the digits of an integer n. By applying the function dsf to 793312220 we can get a loop of length 40.

Example

This is an reiterative process that starts with 7793312220.

Mathematica

dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf, 7793312220, 80]

Prog

(PARI) dsf(n) = my(d = digits(n)); sum(i=1, #d, d[i]^d[i]); \\ Michel Marcus, Apr 21 2014

Crossrefs

Cf. A165942, A166024, A166072, A166121.

Sequence in context: A166121 A348079 A046186 * A104829 A198173 A204499

Adjacent sequences: A166224 A166225 A166226 * A166228 A166229 A166230

Keyword

base,nonn

Author

Ryohei Miyadera, Takuma Nakaoka and Koichiro Nishimura, Oct 09 2009

Extensions

More terms from Michel Marcus, Apr 21 2014

Status

approved