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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). 48
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 (list; graph; refs; listen; history; text; internal format)
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.

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

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), 606-614.

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 *)

CROSSREFS

Cf. A006919, A037271, A037272, A037273, A037275, A037276, A037919-A037941, A048986, A056938.

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,changed

AUTHOR

N. J. A. Sloane, Jeff Burch

EXTENSIONS

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

STATUS

approved

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Last modified September 3 01:21 EDT 2014. Contains 246369 sequences.