A006919 -id:A006919 - 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

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

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 *)

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

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.

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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).

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

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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.

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A037920 Trajectory of 8 under prime factor concatenation procedure.

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

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Comments

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Programs

Mathematica

Python

Extensions