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%I #6 Mar 24 2024 00:31:11
%S 39,1319,211319,12311319,41122319,1431221319,4114232219,2431321319,
%T 2214333119,2231143319,2233311419,2233311419
%N Cycle of the inventory sequence (as in A063850) starting with n consists of prime numbers.
%C It can be proved that any inventory sequence ends in a cycle all of whose terms are <= 10^20.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_207.htm">The Inventory Sequences and Self-Inventoried Numbers</a>
%e The inventory sequence starting with 39 is: 39, 1319, 211319, 12311319, 41122319, 1431221319, 4114232219, 2431321319, 2214333119, 2231143319, 2233311419, 2233311419, .... The cycle is 2233311419, 2233311419, .... and 2233311419 is prime.
%t g[n_] := Module[{seen, r, d, l, i, t}, seen = {}; r = {}; d = IntegerDigits[n]; l = Length[d]; For[i = 1, i <= l, i++, t = d[[i]]; If[ ! MemberQ[seen, t], r = Join[r, IntegerDigits[Count[d, t]]]; r = Join[r, {t}]; seen = Append[seen, t]]]; FromDigits[r]];
%t pr[n_] := Module[{r, t, p1, p, a}, r = {}; t = g[n]; a = True; While[ ! MemberQ[r, t], r = Append[r, t]; t = g[t]]; r = Append[r, t]; p1 = Flatten[Position[r, t]]; p = PrimeQ[Drop[r, p1[[1]]]]; If[MemberQ[p, False], a = False]; a]; l = {}; For[k = 1, k <= 10^4, k++, If[pr[k], l = Append[l, k]]]
%Y Cf. A063850, A078786.
%K base,nice,nonn
%O 1,1
%A _Joseph L. Pe_, Jan 14 2003
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