A166072 Define dsf(n) = A045503(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 decimal digits of n. dsf(809265896) = 808491852 and dsf(808491852) = 437755524,...,dsf(792488396) = 809265896, so these 8 numbers make a loop for the function dsf.

809265896, 808491852, 437755524, 1657004, 873583, 34381154, 16780909, 792488396, 809265896, 808491852, 437755524, 1657004, 873583, 34381154, 16780909, 792488396, 809265896, 808491852, 437755524, 1657004, 873583

Offset: 1

Author

Ryohei Miyadera, Satoshi Hashiba and Koichiro Nishimura, Oct 06 2009

Keywords

• nonn
• base
• easy

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.

Periodic with period 8.

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.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Crossrefs

Cf. A165942, A166024.

Programs

Mathematica

dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf, 809265896,16]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{809265896, 808491852, 437755524, 1657004, 873583, 34381154, 16780909, 792488396},24] (* Ray Chandler, Aug 25 2015 *)

Formula

a(n+1) = dsf(a(n)).

Extensions

Edited by Charles R Greathouse IV, Aug 02 2010

Extended by Ray Chandler, Aug 25 2015