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A066817 Conjectured values of first prime in the orbit f(m), f(f(m)), ..., where f(n) = A067599(n) and m = n-th composite number; or 0 if none exists.

%I A066817 #18 Nov 12 2023 13:27:18
%S A066817 0,2131,23,3224591,0,0,
%T A066817 241127117451117479045190960709721125675426733715695733779133596697360781090711425903130196316185995152974660668512820125356019549490226189398938302252287927928254649608061563193945459975102656949618158919173931,
%U A066817 0,0,0,2251,0,0,0,3224591,314313643123658229739531,97211238048939739899395714118873644859466103898031,0,46747167851021731,3224591,97211238048939739899395714118873644859466103898031,3141114911731,5171
%N A066817 Conjectured values of first prime in the orbit f(m), f(f(m)), ..., where f(n) = A067599(n) and m = n-th composite number; or 0 if none exists.
%C A066817 The terms with 0 value listed above are conjectural. There are no primes < 10^30.
%C A066817 From _Sean A. Irvine_, Nov 09 2023: (Start)
%C A066817 None of the unresolved cases with n < 50 terminates in a prime < 10^130.
%C A066817 Because the trajectories under f can coalesce certain values are known to be equal even if that value is currently unknown. For example, a(1) = a(13) and a(9) = a(14).
%C A066817 Because of the inclusion of exponents 1 in the concatenation defined by f, terms in the trajectory typically grow quicker than in A195264 or A037274.
%C A066817 (End)
%t A066817 (* f returns an array encoding the prime factorization of n *) f[ n_] := Module[ {a, l, i, t = {} }, a = FactorInteger[ n]; l = Length[ a]; For[ i = 1, i <= l, i++, t = Append[ t, a[ [ i]][ [ 1]]]; t = Append[ t, a[ [ i]][ [ 2]]]]; t];
%t A066817 (* g returns the concatenation of the elements of its input array *) g[ x_] := Module[ {r = "", m = Length[ x], l}, For[ l = 1, l <= m, l++, r = StringJoin[ r, ToString[ x[ [ l]]]]]; r];
%t A066817 (* h returns an array of the digits of its input int string *) h[ n_] := IntegerDigits[ ToExpression[ n]]
%t A066817 (* j returns the number formed from the digits in its input array *) j[ x_] := Module[ {r = 0, m = Length[ x], t = x, l}, For[ l = 1, l <= m, l++, r = 10*r + t[ [ 1]]; t = Rest[ t]]; r];
%t A066817 (* k composes the previous functions *) k[ n_] := j[ h[ g[ f[ n]]]]
%t A066817 s[ n_] := Module[ {a=n, r=0}, While[ !PrimeQ[ a] && a<10^30, a=k[ a]]; If[ PrimeQ[ a], r=a]; r]; Table[ s[ i], {i, 2, 50}]
%Y A066817 Cf. A002808, A067599, A067600, A037274, A195264.
%K A066817 nonn,base,less
%O A066817 1,2
%A A066817 _Joseph L. Pe_, Feb 01 2002
%E A066817 Offset changed to 1 by _Jinyuan Wang_, Jul 30 2020
%E A066817 a(7) and a(17) resolved and missing a(21) inserted by _Sean A. Irvine_, Nov 09 2023