

A037274


Home primes: for n >= 2, a(n) = the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached (a(n) = 1 if no prime is ever reached).


54



1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277
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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 103 iterations, a(49) is still composite with 217 digits.
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, 199697, pp. 116119.


LINKS

Table of n, a(n) for n=1..48.
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.
P. De Geest, Home Primes < 100 and Beyond
M. Herman and J. Schiffman, Investigating home primes and their families, Math. Teacher, 107 (No. 8, 2014), 606614.
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.


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


CROSSREFS

Cf. A006919, A037271, A037272, A037273, A037275, A037276, A037919A037941, A048986, A056938.
Cf. A195264 (use exponents instead of repeating primes).
Cf. A084318 (use only one copy of each prime), A248713 (FermiDirac analog: use unique representation of n>1 as a product of distinct terms of A050376).
Cf. also A120716 and related sequences.
Sequence in context: A160759 A191835 A195264 * A037275 A142960 A117324
Adjacent sequences: A037271 A037272 A037273 * A037275 A037276 A037277


KEYWORD

nonn,nice,base


AUTHOR

N. J. A. Sloane, Jeff Burch


EXTENSIONS

Corrected and extended by Karl W. Heuer, Sep 30 2003


STATUS

approved



