A189398 a(n) = 2^d(1) * 3^d(2) * ... * prime(k)^d(k), where d(1)d(2)...d(k) is the decimal representation of n.

2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 16, 48, 144, 432, 1296, 3888, 11664, 34992, 104976, 314928, 32

Offset: 1

Author

Reinhard Zumkeller, May 03 2011

Keywords

• nonn
• base
• look

Comments

Not the same as A061509: a(n) = A061509(n) for n <= 100; a(101)=2^1*3^0*5^1=10 <> A061509(101)=2^1*3^1=6;

a(A052382(n)) = A000079(A000030(a052382(n))) = A061509(A052382(n));

a(A002275(n)) = A002110(n): a(n-th rep-unit) = n-th primorial;

a(n*A011557(k)) = a(n): trailing zeros don't matter;

A001221(a(n)) = A055640(n): number of distinct prime factors of a(n) = number of nonzero digits of n;

A001222(a(n)) = A007953(n): number of all prime factors of a(n) = sum of digits of n;

a(81312000) = 2^8*3^1*5^3*7^1*11^2*13^0*17^0*19^0 = 81312000, the smallest fixed point, is called the Meertens number.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Richard S. Bird, Functional Pearl: Meertens number, Journal of Functional Programming 8 (1), Jan 1998, 83-88.
• Wikipedia, Meertens number

Crossrefs

Cf. A000040.

Programs

Haskell

import Data.Char (digitToInt)
import Data.List (findIndices)
a189398 n = product $ zipWith (^) a000040_list (map digitToInt $ show n)
-- Two computations of the Meertens number: the first is brute force,
meertens = map succ $ findIndices (\x -> a189398 x == x) [1..]
-- ... and the second is more efficient, from Bird reference, page 87:
meertens' k = [n | (n,g) <- candidates (0,1), n == g] where
  candidates        = concat . map (search pps) . tail . labels ps
  ps : pps          = map (\p -> iterate (p *) 1) $ take k a000040_list
  search [] x       = [x]
  search (ps:pps) x = x : concat (map (search pps) (labels ps x))
  labels ps (n,g)   = zip (map (10*n +) [0..9]) (chop $ map (g *) ps)
  chop              = takeWhile (< 10^k)
-- Time and space required, GHC interpreted, Mac OS X, 2.66 GHz:
-- for >head meertens: (466.87 secs, 254780027728 bytes);
-- for >meertens' 8:   (  0.28 secs,     62027124 bytes).

Maple

a:= n-> `if`(n=0, 1, ithprime(length(n))^irem(n, 10, 'm') *a(m)):
seq(a(n), n=1..110);  # Alois P. Heinz, May 04 2011

Mathematica

a[n_] := (p = Prime[Range[Length[d = IntegerDigits[n]]]]; Times @@ (p^d)); Array[a, 50] (* Jean-François Alcover, Jan 09 2016 *)

PARI

a(n)=my(d=digits(n),p=primes(#d)); prod(i=1,#d,p[i]^d[i]) \\ Charles R Greathouse IV, Aug 19 2014

Python

from sympy import prime
from operator import mul
from functools import reduce
def A189398(n):
    return reduce(mul, (prime(i)**int(d) for i,d in enumerate(str(n),start=1)))
# Chai Wah Wu, Aug 31 2014

Python

# implementation using recursion
from sympy import prime
def _A189398(n):
    nlen = len(n)
    return _A189398(n[:-1])*prime(nlen)**int(n[-1]) if nlen > 1 else 2**int(n)
def A189398(n):
    return _A189398(str(n))
# Chai Wah Wu, Aug 31 2014