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

Author

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

Keywords

• base
• nonn

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.

Examples

			This is an reiterative process that starts with 7793312220.
		

Links

• 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.

Crossrefs

Cf. A165942, A166024, A166072, A166121.

Programs

Mathematica

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

PARI

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

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.

Extensions

More terms from Michel Marcus, Apr 21 2014