A048986 Home primes in base 2: primes reached when you start with n and (working in base 2) concatenate its prime factors (A048985); repeat until a prime is reached (or -1 if no prime is ever reached). Answer is written in base 10.

1, 2, 3, 31, 5, 11, 7, 179, 29, 31, 11, 43, 13, 23, 29, 12007, 17, 47, 19, 251, 31, 43, 23, 499, 4091, 4091, 127, 4091, 29, 127, 31, 1564237, 59, 4079, 47, 367, 37, 83, 61, 383, 41, 179, 43, 499, 4091, 4091, 47, 683, 127, 173, 113, 173, 53, 191, 4091

Offset: 1

Author

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

Keywords

• nonn
• base
• nice

Comments

a(1) = 1 by convention.

The first binary home prime that is not known is a(2295). - Ely Golden, Jan 09 2017

Examples

			4 = 2*2 -> 1010 = 10 = 2*5 ->10101 = 21 = 3*7 -> 11111 = 31 = prime.
		

Links

• Ely Golden, Table of n, a(n) for n = 1..2294
• Patrick De Geest, Home Primes
• Ely Golden, Mersenne Wiki Home Primes base 2
• Ely Golden, Table of n, a(n) for n = 1..3000 (a-file)

Crossrefs

Cf. A048985, A037274, A049065.

Programs

Mathematica

f[n_] := Module[{fi}, If[PrimeQ[n], n, fi = FactorInteger[n]; Table[ First[#], {Last[#]}]& /@ fi // Flatten // IntegerDigits[#, 2]& // Flatten // FromDigits[#, 2]&]]; a[1] = 1; a[n_] := TimeConstrained[FixedPoint[f, n], 1] /. $Aborted -> -1; Array[a, 55] (* Jean-François Alcover, Jan 01 2016 *)

Python

from sympy import factorint, isprime
def f(n):
    if n == 1: return 1
    return int("".join(bin(p)[2:]*e for p, e in factorint(n).items()), 2)
def a(n):
    if n == 1: return 1
    while not isprime(n): n = f(n)
    return n
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 07 2022

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 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
radix=2
index=2
while(index<=1344):
    print(str(index)+" "+str(hp(index,radix)))
    index+=1