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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). 60
 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 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. 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. 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. 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 xrange(len(f)):         for d in xrange(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)) CROSSREFS Cf. A006919, A037271, A037272, A037273, A037275, A037276, A037919-A037941, A048986, A056938. Cf. A195264 (use exponents instead of repeating primes). Cf. A084318 (use only one copy of each prime), A248713 (Fermi-Dirac analog: use unique representation of n>1 as a product of distinct terms of A050376). Cf. also A120716 and related sequences. Sequence in context: A317550 A195264 A316941 * A321225 A037275 A142960 Adjacent sequences:  A037271 A037272 A037273 * A037275 A037276 A037277 KEYWORD nonn,nice,base AUTHOR EXTENSIONS Corrected and extended by Karl W. Heuer, Sep 30 2003 STATUS approved

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Last modified November 15 07:25 EST 2018. Contains 317225 sequences. (Running on oeis4.)