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.

0, 2131, 23, 3224591, 0, 0, 241127117451117479045190960709721125675426733715695733779133596697360781090711425903130196316185995152974660668512820125356019549490226189398938302252287927928254649608061563193945459975102656949618158919173931, 0, 0, 0, 2251, 0, 0, 0, 3224591, 314313643123658229739531, 97211238048939739899395714118873644859466103898031, 0, 46747167851021731, 3224591, 97211238048939739899395714118873644859466103898031, 3141114911731, 5171

Offset: 1

Author

Joseph L. Pe, Feb 01 2002

Keywords

• nonn
• base
• less

Comments

The terms with 0 value listed above are conjectural. There are no primes < 10^30.

From Sean A. Irvine, Nov 09 2023: (Start)

None of the unresolved cases with n < 50 terminates in a prime < 10^130.

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).

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.

(End)

Crossrefs

Cf. A002808, A067599, A067600, A037274, A195264.

Programs

Mathematica

(* 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];
(* 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];
(* h returns an array of the digits of its input int string *) h[ n_] := IntegerDigits[ ToExpression[ n]]
(* 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];
(* k composes the previous functions *) k[ n_] := j[ h[ g[ f[ n]]]]
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}]

Extensions

Offset changed to 1 by Jinyuan Wang, Jul 30 2020

a(7) and a(17) resolved and missing a(21) inserted by Sean A. Irvine, Nov 09 2023