OFFSET
1,2
COMMENTS
The initial 1 could have been omitted.
Probabilistic arguments give exactly zero for the chance that the sequence of integers starting at n contains no prime, the expected number of primes being given by a divergent sequence. - J. H. Conway
After over 100 iterations, a(49) is still composite - see A056938 for the latest information.
More terms:
a(50) to a(60) are 3517, 317, 2213, 53, 2333, 773, 37463, 1129, 229, 59, 35149;
a(61) to a(65) are 61, 31237, 337, 1272505013723, 1381321118321175157763339900357651;
a(66) to a(76) are 2311, 67, 3739, 33191, 257, 71, 1119179, 73, 379, 571, 333271.
This is different from A195264. Here 8 = 2^3 -> 222 -> ... -> 3331113965338635107 (a prime), whereas in A195264 8 = 2^3 -> 23 (a prime). - N. J. A. Sloane, Oct 12 2014
REFERENCES
Jeffrey Heleen, Family Numbers: Mathemagical Black Holes, Recreational and Educational Computing, 5:5, pp. 6, 1990.
Jeffrey Heleen, Family numbers: Constructing Primes by Prime Factor Splicing, J. Recreational Math., Vol. 28 #2, 1996-97, pp. 116-119.
LINKS
Christian N. K. Anderson, Table of known values of n, # of steps to reach a(n), and a(n) or NA if a(n) has 30 digits or more. Also, the trajectory, with factors separated by a |, terminated by either "(end)" or "-> ?" if a(n) has 30 digits or more.
Patrick De Geest, Home Primes < 100 and Beyond
M. Herman and J. Schiffman, Investigating home primes and their families, Math. Teacher, 107 (No. 8, 2014), 606-614.
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
Eric Weisstein's World of Mathematics, Home Prime.
Wikipedia, Home prime
EXAMPLE
9 = 3*3 -> 33 = 3*11 -> 311, prime, so a(9) = 311.
The trajectory of 8 is more interesting:
8 ->
2 * 2 * 2 ->
2 * 3 * 37 ->
3 * 19 * 41 ->
3 * 3 * 3 * 7 * 13 * 13 ->
3 * 11123771 ->
7 * 149 * 317 * 941 ->
229 * 31219729 ->
11 * 2084656339 ->
3 * 347 * 911 * 118189 ->
11 * 613 * 496501723 ->
97 * 130517 * 917327 ->
53 * 1832651281459 ->
3 * 3 * 3 * 11 * 139 * 653 * 3863 * 5107
and 3331113965338635107 is prime, so a(8) = 3331113965338635107.
MAPLE
b:= n-> parse(cat(sort(map(i-> i[1]$i[2], ifactors(n)[2]))[])):
a:= n-> `if`(isprime(n) or n=1, n, a(b(n))):
seq(a(n), n=1..48); # Alois P. Heinz, Jan 09 2021
MATHEMATICA
f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], { #[[2]] }] & /@ FactorInteger@n, 2]; g[n_] := NestWhile[ f@# &, n, !PrimeQ@# &]; g[1] = 1; Array[g, 41] (* Robert G. Wilson v, Sep 22 2007 *)
PROG
(PARI) step(n)=my(f=factor(n), s=""); for(i=1, #f~, for(j=1, f[i, 2], s=Str(s, f[i, 1]))); eval(s)
a(n)=if(n<4, return(n)); while(!isprime(n), n=step(n)); n \\ Charles R Greathouse IV, May 14 2015
(SageMath)
def digitLen(x, n):
r=0
while(x>0):
x//=n
r+=1
return r
def concatPf(x, n):
r=0
f=list(factor(x))
for c in range(len(f)):
for d in range(f[c][1]):
r*=(n**digitLen(f[c][0], n))
r+=f[c][0]
return r
def hp(x, n):
x1=concatPf(x, n)
while(x1!=x):
x=x1
x1=concatPf(x1, n)
return x
#example: prints the home prime of 8 in base 10
print(hp(8, 10))
(Python)
from sympy import factorint, isprime
def f(n): return int("".join(str(p)*e for p, e in factorint(n).items()))
def a(n):
if n == 1: return 1
fn = n
while not isprime(fn): fn = f(fn)
return fn
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Jul 11 2022
CROSSREFS
KEYWORD
nonn,nice,base,changed
AUTHOR
EXTENSIONS
Corrected and extended by Karl W. Heuer, Sep 30 2003
STATUS
approved