A133500 The powertrain or power train map: Powertrain(n): if abcd... is the decimal expansion of a number n, then the powertrain of n is the number n' = a^b*c^d* ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 1, 6, 36, 216, 1296

Offset

0,3

Comments

We take 0^0 = 1.

The fixed points are in A135385.

For 1-digit or 2-digit numbers this is the same as A075877. - R. J. Mathar, Mar 28 2012

a(A221221(n)) = A133048(A221221(n)) = A222493(n). - Reinhard Zumkeller, May 27 2013

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

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)

Example

20 -> 2^0 = 1,
21 -> 2^1 = 2,
24 -> 2^4 = 16,
39 -> 3^9 = 19683,
623 -> 6^2*3 = 108,
etc.

Maple

powertrain:=proc(n) local a, i, n1, n2, t1, t2; n1:=abs(n); n2:=sign(n); t1:=convert(n1, base, 10); t2:=nops(t1); a:=1; for i from 0 to floor(t2/2)-1 do a := a*t1[t2-2*i]^t1[t2-2*i-1]; od: if t2 mod 2 = 1 then a:=a*t1[1]; fi; RETURN(n2*a); end; # N. J. A. Sloane, Dec 03 2007

Mathematica

ptm[n_]:=Module[{idn=IntegerDigits[n]}, If[EvenQ[Length[idn]], Times@@( #[[1]]^ #[[2]] &/@Partition[idn, 2]), (Times@@(#[[1]]^#[[2]] &/@ Partition[ Most[idn], 2]))Last[idn]]]; Array[ptm, 70, 0] (* Harvey P. Dale, Jul 15 2019 *)

Prog

(Haskell)
a133500 = train . reverse . a031298_row where
   train []       = 1
   train [x]      = x
   train (u:v:ws) = u ^ v * (train ws)
-- Reinhard Zumkeller, May 27 2013
(Python)
def A133500(n):
    s = str(n)
    l = len(s)
    m = int(s[-1]) if l % 2 else 1
    for i in range(0, l-1, 2):
        m *= int(s[i])**int(s[i+1])
    return m # Chai Wah Wu, Jun 16 2017

Crossrefs

Cf. A075877, A133501 (number of steps to reach fixed point), A133502, A135385 (the conjectured list of fixed points), A135384 (numbers which converge to 2592). For records see A133504, A133505; for the fixed points that are reached when this map is iterated starting at n, see A287877.

Cf. also A133048 (powerback), A031346 and A003001 (persistence).

Cf. also A031298, A007376.

Sequence in context: A175400 A175399 A075877 * A256229 A052423 A126616

Adjacent sequences: A133497 A133498 A133499 * A133501 A133502 A133503

Keyword

nonn,base,look

Author

J. H. Conway, Dec 03 2007

Status

approved