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.

-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

Author

N. J. A. Sloane, Jeff Burch

Keywords

• sign
• nice
• hard
• base
• more

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A010051, A027746.

Programs

Haskell

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

Mathematica

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 *)

PARI

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

Python

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

Extensions

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