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A037274
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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).
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42
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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; internal format)
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OFFSET
| 1,2
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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 - John Conway (conway(AT)math.princeton.edu)
After 103 iterations, a(49) is still composite with 217 digits.
a(50:60)=3517,317,2213,53,2333,773,37463,1129,229,59,35149
a(61:65)=61,31237,337,1272505013723,1381321118321175157763339900357651
a(66:76)=2311,67,3739,33191,257,71,1119179,73,379,571,333271
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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.
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LINKS
| P. De Geest, Home Primes < 100 and Beyond
Eric Weisstein's World of Mathematics, Home Prime.
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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.
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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 (rgwv(AT)rgwv.com), Sep 22 2007 *)
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CROSSREFS
| Cf. A006919, A037271-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
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KEYWORD
| nonn,nice,base
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jeff Burch (gburch(AT)erols.com)
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EXTENSIONS
| Corrected and extended by Karl W. Heuer, Sep 30 2003
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