A056938 -id:A056938 - 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

A037273 Number of steps to reach a prime under "replace n with concatenation of its prime factors", or -1 if no prime is ever reached.

Original entry on oeis.org

-1, 0, 0, 2, 0, 1, 0, 13, 2, 4, 0, 1, 0, 5, 4, 4, 0, 1, 0, 15, 1, 1, 0, 2, 3, 4, 4, 1, 0, 2, 0, 2, 1, 5, 3, 2, 0, 2, 1, 9, 0, 2, 0, 9, 6, 1, 0, 15

Offset: 1

N. J. A. Sloane, Jeff Burch

Starting with 49, no prime has been reached after 79 steps.

a(49) > 118, see A056938 and FactorDB link. - Michael S. Branicky, Nov 19 2020

			13 is already prime, so a(13) = 0.
Starting with 14 we get 14 = 2*7, 27 = 3*3*3, 333 = 3*3*37, 3337 = 47*71, 4771 = 13*367, 13367 is prime; so a(14) = 5.

Patrick De Geest, Home Primes
FactorDB, Home prime base 10
Eric Weisstein's World of Mathematics, Home Prime

Cf. A037271, A037272, A037274, A037275, A056938.

Cf. A010051, A027746.

a037273 1 = -1
a037273 n = length $ takeWhile ((== 0) . a010051) $
   iterate (\x -> read $ concatMap show $ a027746_row x :: Integer) n
-- Reinhard Zumkeller, Jan 08 2013

nxt[n_] := FromDigits[Flatten[IntegerDigits/@Table[#[[1]], {#[[2]]}]&/@ FactorInteger[n]]]; Table[Length[NestWhileList[nxt, n, !PrimeQ[#]&]] - 1, {n, 48}] (* Harvey P. Dale, Jan 03 2013 *)

row_a027746(n, o=[1])=if(n>1, concat(apply(t->vector(t[2], i, t[1]), Vec(factor(n)~))), o) \\ after M. F. Hasler in A027746
tonum(vec) = my(s=""); for(k=1, #vec, s=concat(s, Str(vec[k]))); eval(s)
a(n) = if(n==1, return(-1)); my(x=n, i=0); while(1, if(ispseudoprime(x), return(i)); x=tonum(row_a027746(x)); i++) \\ Felix Fröhlich, May 17 2021

from sympy import factorint
def a(n):
    if n < 2: return -1
    klst, f = [n], sorted(factorint(n, multiple=True))
    while len(f) > 1:
        klst.append(int("".join(map(str, f))))
        f = sorted(factorint(klst[-1], multiple=True))
    return len(klst) - 1
print([a(n) for n in range(1, 49)]) # Michael S. Branicky, Aug 02 2021

Edited by Charles R Greathouse IV, Apr 23 2010
a(1) = -1 by Reinhard Zumkeller, Jan 08 2013
Name edited by Felix Fröhlich, May 17 2021

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 *)

A084319 Orbit of 91 under function described in A084317.

Original entry on oeis.org

91, 713, 2331, 3737, 37101, 383149, 1329473, 10912197, 328312853, 1129846623, 3735159117, 31245053039, 173977184859, 3293176308321, 319269241788861, 371325123869195203, 1278647733810375857, 1665622037676698019, 31742715741254857303, 56627509560552923867

Offset: 0

Labos Elemer, Jun 16 2003

This sequence takes 64 steps to reach a prime (which implies A343156(91)=64). - N. J. A. Sloane, Apr 07 2021

			a(29) = 19797186041838357425713338412621, the 29th iterate.

Eric Angelini, W. Edwin Clark, Hans Havermann, Frank Stevenson, Allan C. Wechsler, and others, Postings to Math Fun mailing list, April 2021.

W. Edwin Clark, Table of n, a(n) for n = 0..63 [The trajectory to the first prime.]

Cf. A084317, A056938, A084318, A343156.

NestList[FromDigits@ Flatten@ IntegerDigits@ Map[First, FactorInteger@ #] &, 91, 17] (* Michael De Vlieger, Mar 25 2017 *)

Example corrected by Rémy Sigrist, Apr 07 2021

A006919 Write down all the prime divisors in previous term.

Original entry on oeis.org

8, 222, 2337, 31941, 33371313, 311123771, 7149317941, 22931219729, 112084656339, 3347911118189, 11613496501723, 97130517917327, 531832651281459, 3331113965338635107, 3331113965338635107

Offset: 1

N. J. A. Sloane

H. Jaleebi, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Patrick De Geest, Home Primes

Cf. A056938 (same for a(1)=49), A037271-A037276, A048985, A048986, A049065.

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, 8, 15 ]
NestList[FromDigits[Flatten[IntegerDigits/@(Table[First[#],{Last[#]}]& /@ FactorInteger[#])]]&,8,15] (* Harvey P. Dale, Dec 04 2011 *)

first(N, a=8)=vector(N,i,if(i>1,a=A037276(a),a)) \\ M. F. Hasler, Oct 07 2022

a(n+1) = A037276(a(n)), a(1) = 8. - M. F. Hasler, Oct 07 2022

More terms from Robert G. Wilson v, Sep 05 2000, who remarks that sequence stabilizes at 13th term with a prime.

A049065 Record primes reached in A048986.

Original entry on oeis.org

2, 3, 31, 179, 12007, 1564237, 17320726789571, 401278664296369, 576312045441408907, 37246812772043701411753149215934377, 3690727229000499480592573891534356177653018575120050845976045596834749951228879

Offset: 1

Michael B Greenwald (mbgreen(AT)central.cis.upenn.edu)

The value 37246812772043701411753149215934377 is the base-2 home prime for 922 and occurs after 66 steps. The value 3690727229000499480592573891534356177653018575120050845976045596834749951228879 is the base-2 home prime for 1345 and occurs after 131 steps. The next term (home prime for 2295) contains at least 124 digits. Computation of further terms involves large factorizations. - Sean A. Irvine, Aug 04 2005 [corrected Jul 17 2021]

Patrick De Geest, Home Primes

Cf. A056938, A037271-A037276, A048985, A048986, A049065.

a(10)-a(11) from Sean A. Irvine, Aug 04 2005

a(10) corrected by Sean A. Irvine, Jul 17 2021

A238579 Home prime sequence (see A037274) starting at 146.

Original entry on oeis.org

146, 273, 3713, 4779, 333359, 1325643, 3111717139, 29177760383, 69142225413, 3471134339561, 7980350584141, 1324115101168677, 33147123900129853, 1941131324815763997, 37816317113233982621, 291304010934939102849, 333777134924210136703397, 7409854792211363875345439

Offset: 1

J. Lowell, Mar 30 2014

Second sequence of this kind with as yet unknown trajectory (see A037274, A056938). - J. Lowell, Apr 27 2017

			146 = 2*73 so next term is 273; 273 = 3*7*13 so next term is 3713.

Sean A. Irvine, Table of n, a(n) for n = 1..100
Wikipedia, Home prime

Cf. A037274 (home primes), A056938.

NestList[FromDigits@ Flatten@ Map[IntegerDigits, FactorInteger[#] /. {p_, e_} /; p >= 1 :> If[p == 1, 1, ConstantArray[p, e]]] &, 146, 17] (* Michael De Vlieger, Apr 27 2017 *)

lista(nn) = {n = 146; for (i=1 , nn, print1(n, ", "); f = factor(n); s = ""; for (j=1, #f~, for (k=1, f[j, 2], s = concat(s, Str(f[j, 1])););); n = eval(s););} \\ Michel Marcus, Mar 31 2014

One more term added by J. Lowell, Mar 30 2014
More terms from Jon E. Schoenfield, Mar 30 2014
One further term from N. J. A. Sloane, Mar 31 2014

A331603 a(1) = 1; for n > 1, if a(n-1) is composite then a(n) is the concatenation of all the prime factors in order of a(n-1), otherwise a(n) is the smallest number not yet appearing in the sequence.

Original entry on oeis.org

1, 2, 3, 4, 22, 211, 5, 6, 23, 7, 8, 222, 2337, 31941, 33371313, 311123771, 7149317941, 22931219729, 112084656339, 3347911118189, 11613496501723, 97130517917327, 531832651281459, 3331113965338635107, 9, 33, 311, 10, 25, 55, 511, 773, 11, 12, 223, 13, 14, 27, 333, 3337, 4771, 13367, 15

Offset: 1

Scott R. Shannon, Jan 21 2020

Assuming that all numbers when replaced with the concatenation of their prime factors will eventually reach a prime (see A037274), this sequence will contain all positive integers. a(158) = 49 which currently has no known 'home prime' in the iterative sequence of prime factor replacements; see A056938.

			a(5) = 22 as a(4) = 4 which has a factorization 4 = 2*2, so the concatenation of factors is '22'.
a(7) = 5 as a(6) = 211 which is prime, and 5 is the smallest number not yet appearing in the sequence.
a(14) = 31941 as a(13) = 2337 which has a factorization 2337 = 3*19*41, so the concatenation of factors is '31941'.

Michael De Vlieger, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Home Prime.

Cf. A000040, A027746, A037274, A037271, A006919, A056938.

nn = 43; c[] = 0; a[1] = c[1] = u = 1; While[c[u] > 0, u++]; Do[If[CompositeQ[#], k = FromDigits@ Flatten@ Map[IntegerDigits[#] &, ConstantArray[##] & @@@ FactorInteger[#]], k = u] &@ a[i - 1]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 12 2022 *)

A343157 Trajectory of 407 under the map x -> A084317(x).

Original entry on oeis.org

407, 1137, 3379, 31109, 132393, 344131, 1731653, 71143523, 115771019, 7133141039, 18152375353, 723112747673, 1938058565667, 372411163329269, 646991575604859, 3500960117162747, 19920988418382133, 479222853318661919, 3877130279948783893, 71942196909541476259, 7170749184914732550379

Offset: 0

Hans Havermann, Apr 07 2021

It is not known if any a(n) is a prime (see discussion in A343156). - N. J. A. Sloane, Apr 07 2021

Eric Angelini, W. Edwin Clark, Hans Havermann, Frank Stevenson, Allan C. Wechsler, and others, Postings to Math Fun mailing list, April 2021.

Jinyuan Wang, Table of n, a(n) for n = 0..91 (terms 0..84 from Hans Havermann)

Cf. A056938, A084317, A343156.

A063970 a(1) = 2; for n>1, write down all divisors of the previous term in order of magnitude.

Original entry on oeis.org

2, 12, 1234612, 1247915831639077814156283086536173061234612

Offset: 1

Labos Elemer, Sep 05 2001

The next term has 3104 digits. - Harvey P. Dale, May 28 2017

			Divisors of a(3)={1, 2, 4, 79, 158, 316, 3907, 7814, 15628, 308653, 617306, 1234612}

Cf. A006919, A056938.

NestList[FromDigits[Flatten[IntegerDigits/@Divisors[#]]]&,2,4] (* Harvey P. Dale, May 28 2017 *)

from sympy import divisors
def aupton(terms):
  alst = [2]
  for n in range(2, terms+1):
    alst.append(int("".join(str(d) for d in divisors(alst[-1]))))
  return alst
print(aupton(4)) # Michael S. Branicky, Feb 12 2021

Next term has more than 3000 decimal digits.

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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A037273 Number of steps to reach a prime under "replace n with concatenation of its prime factors", or -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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A084319 Orbit of 91 under function described in A084317.

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A006919 Write down all the prime divisors in previous term.

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A049065 Record primes reached in A048986.

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A238579 Home prime sequence (see A037274) starting at 146.

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