A166383 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(1583236420) =1682731 and dsf(1682731) = 18470991,...,dsf(388290999) = 1583236420,.. in this way this 97 numbers make a loop for the function dsf. In fact this is the longest loop for dsf function in the set of all nonnegative integers.

1583236420, 16827317, 18470991, 792441996, 1163132183, 16823961, 404291050, 387424134, 17601586, 17697199, 1163955211, 387473430, 18424896, 421022094, 387421016, 17647705, 2520668, 16873662, 17740759, 389894501, 808398820

Offset: 1

Author

Ryohei Miyadera, Oct 13 2009

Keywords

• nonn
• base

Comments

In fact there are only 8 loops in the whole nonnegative integers for the dsf-function that we defined. We have discovered this fact with the calculation by Mathematica and other general purpose languages. We have presented 6 loops to this On-Line Encyclopedia of Integer Sequences, and other two loops are in fact fixed points {1} and {3435}. It is easy to see that dsf(1) = 1 and dsf(3435) = 3^3+4^4+3^3+5^5=3435.

Examples

			This is an iterative process that starts with 1583236420.
		

Links

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

Crossrefs

Cf. A165942, A166024, A166072, A166121, A166227.

Programs

Mathematica

dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf,1583236420,194]

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 1583236420 repeatedly we can get a loop of the length of 97.