A037273 -id:A037273 - cp's OEIS frontend

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).

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

Offset: 1

N. J. A. Sloane, Jeff Burch

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

			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.

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.

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

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.

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

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 *)

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

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

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))

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

A037271 Number of steps to reach a prime under "replace n with concatenation of its prime factors" when applied to n-th composite number, or -1 if no such number exists.

Original entry on oeis.org

2, 1, 13, 2, 4, 1, 5, 4, 4, 1, 15, 1, 1, 2, 3, 4, 4, 1, 2, 2, 1, 5, 3, 2, 2, 1, 9, 2, 9, 6, 1, 15

Offset: 1

Jeff Burch

a(33) is presently unknown: starting with 49, no prime has been reached after 110 steps. See A037274 for the latest information.

			Starting with 14 (the seventh composite number) we get 14=2*7, 27=3*3*3, 333=3*3*37, 3337=47*71, 4771=13*367, 13367 is prime; so a(7)=5.

Patrick De Geest, Home Primes
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

Cf. A037272, A037273, A037274, A056938, A002808, A037276, A027746, A230305.

a037271 = length . takeWhile ((== 0) . a010051'') .
                             iterate a037276 . a002808
-- Reinhard Zumkeller, Apr 03 2012

maxComposite = 49; maxIter = 40; concat[n_] := FromDigits[ Flatten[ IntegerDigits /@ Flatten[ Apply[ Table, {#[[1]], {#[[2]]}} & /@ FactorInteger[n], {1}]]]]; composites = Select[ Range[2, maxComposite], ! PrimeQ[#] &]; a[n_] := ( lst = NestWhileList[ concat, composites[[n]], ! PrimeQ[#] &, 1, maxIter]; If[PrimeQ[ Last[lst]], Length[lst] - 1, - 1]); Table[a[n], {n, 1, Length[composites]}] (* Jean-François Alcover, Jul 10 2012 *)

A056938 Concatenate all the prime divisors in previous term (with repetition), starting at 49.

Original entry on oeis.org

49, 77, 711, 3379, 31109, 132393, 344131, 1731653, 71143523, 11115771019, 31135742029, 717261644891, 11193431873899, 116134799345907, 3204751189066719, 31068250396355573, 62161149980213343, 336906794442245927, 734615161567701999, 31318836286194043641

Offset: 1

Robert G. Wilson v, Sep 05 2000

This sequence provides a record of the search for the home prime for 49.

This sequence has now been followed for 117 steps without a prime being reached (after which of course it would simply repeat).

Patrick De Geest, Table of n, a(n) for n = 1..119
P. De Geest, Home Primes
Eric Weisstein's World of Mathematics, Home Prime

Cf. A006919, A037273, A037274, A037271, A238579.

g[n_] := (x = n; d = {}; While[FactorInteger[x] != {}, f = FactorInteger[x, FactorComplete -> True][[1, 1]]; x = x/f; AppendTo[d, IntegerDigits[f]]]; FromDigits[Flatten[d]]); NestList[g, 49, 25]
(* Second program: *)
NestList[FromDigits@ Flatten@ Map[IntegerDigits, FactorInteger[#] /. {p_, e_} /; p >= 1 :> If[p == 1, 1, ConstantArray[p, e]]] &, 49, 16] (* Michael De Vlieger, Apr 27 2017 *)

a=vector(35); a[1]=49; for(k=2,length(a), f=factor(a[k-1]); for(i=1,matsize(f)[1], l=10^ceil(log(f[i,1])/log(10)); for(j=1,f[i,2], a[k]=a[k]*l+f[i,1]))) \\ M. F. Hasler, Mar 09 2007

b-file updated by Max Alekseyev, Nov 28 2017

A037272 Primes reached in A037271, or -1 if no such prime exists.

Original entry on oeis.org

211, 23, 3331113965338635107, 311, 773, 223, 13367, 1129, 31636373, 233, 3318308475676071413, 37, 211, 331319, 773, 3251, 13367, 227, 547, 241271, 311, 31397, 1129, 71129, 373, 313, 3314192745739, 379, 22815088913, 3411949, 223, 6161791591356884791277

Offset: 1

N. J. A. Sloane, Jeff Burch

The next term, a(33), is presently unknown. See A037274. - N. J. A. Sloane, Apr 30 2014

P. De Geest, Home Primes
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.

Cf. A037271, A037273, A037274.

A037941 Trajectory of 48 under prime factor concatenation procedure.

Original entry on oeis.org

48, 2, 2, 2, 2, 3, 71, 313, 3, 11, 2161, 3, 13, 199, 401, 19, 43, 109, 3517, 11, 17, 109, 877, 1087, 23, 1481, 7039, 46591, 3, 3, 7, 53, 67, 1034726207, 3, 11251223678242069, 23, 4583, 2952795526741, 359, 5782291, 1130063089

Offset: 0

Jeff Burch

			48 = 2*2*2*2*3, so the next terms are 2,2,2,2,3; 22223 = 71*313, so the next terms are 71, 313; 71313 = 3*11*2161, so then we get 3, 11, 2161; etc.

Sean A. Irvine, Table of n, a(n) for n = 0..54
Patrick De Geest, Home Primes

Cf. A037273.

Edited by Charles R Greathouse IV, Apr 30 2010

A037919 Trajectory of 4 under prime factor concatenation procedure.

Original entry on oeis.org

4, 2, 2, 2, 11, 211

Offset: 0

Jeff Burch

			May be regarded as the rows of an irregular triangle:
4
2 2
2 11
211. - _N. J. A. Sloane_, Jun 15 2022

Patrick De Geest, Home Primes

Cf. A037273.

Edited by Charles R Greathouse IV, Apr 30 2010

A037299 Prime substrings of prime numbers in A037272.

Original entry on oeis.org

2, 11, 2, 3, 3, 3, 3, 11, 139, 653, 3863, 5107, 3, 11, 7, 73, 2, 2, 3, 13, 367, 11, 29, 3, 163, 6373, 2, 3, 3, 3, 31, 8308475676071413, 3, 7, 2, 11, 3, 3, 13, 19, 7, 73, 3, 251, 13, 367, 2, 2, 7, 5, 47, 2, 41, 271, 3, 11, 3, 13, 97, 11, 29, 7, 11, 29, 3, 73, 3, 13, 3, 31, 41

Offset: 0

Jeff Burch

nonn

There should be an analogous sequence for A037273 (N. J. A. Sloane).

A037300 Number of prime substrings of prime numbers in A037272.

Original entry on oeis.org

2, 2, 8, 2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 4, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 4, 2, 2, 3, 2, 4

Offset: 0

Jeff Burch

nonn

There should be an analogous sequence for A037273 (N. J. A. Sloane).

A037920 Trajectory of 8 under prime factor concatenation procedure.

Original entry on oeis.org

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

Offset: 0

Jeff Burch

Patrick De Geest, Home Primes

Cf. A037273, A006919.

from sympy import factorint
def iterate(n):
    flst, f = [n], sorted(factorint(n, multiple=True))
    while len(f) > 1:
        flst += f
        f = sorted(factorint(int("".join(map(str, f))), multiple=True))
    return flst
print(iterate(8)) # Michael S. Branicky, Aug 02 2021

Edited by Charles R Greathouse IV, Apr 30 2010

A037921 Trajectory of 9 under prime factor concatenation procedure.

Original entry on oeis.org

9, 3, 3, 3, 11, 311

Offset: 1

Jeff Burch

			9 -> 3,3 -> 3,11 -> 311

P. De Geest, Home Primes

Cf. A037274, A037273.

Edited by Charles R Greathouse IV, Apr 30 2010

cp's OEIS Frontend

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).

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A037271 Number of steps to reach a prime under "replace n with concatenation of its prime factors" when applied to n-th composite number, or -1 if no such number exists.

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A056938 Concatenate all the prime divisors in previous term (with repetition), starting at 49.

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A037272 Primes reached in A037271, or -1 if no such prime exists.

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A037941 Trajectory of 48 under prime factor concatenation procedure.

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A037919 Trajectory of 4 under prime factor concatenation procedure.

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A037299 Prime substrings of prime numbers in A037272.

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A037300 Number of prime substrings of prime numbers in A037272.

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A037920 Trajectory of 8 under prime factor concatenation procedure.

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