A222493 -id:A222493 - cp's OEIS frontend

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^bc^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

J. H. Conway, Dec 03 2007

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

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

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)

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.

a133500 = train . reverse . a031298_row where
   train []       = 1
   train [x]      = x
   train (u:v:ws) = u ^ v * (train ws)
-- Reinhard Zumkeller, May 27 2013

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

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 *)

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

A133048 Powerback(n): reverse the decimal expansion of n, drop any leading zeros, then apply the powertrain map of A133500 to the resulting number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 1, 4, 9, 16, 25, 36, 49, 64, 81, 3, 1, 8, 27, 64, 125, 216, 343, 512, 729, 4, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 5, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 6, 1, 64, 729, 4096, 15625, 46656, 117649

Offset: 0

J. H. Conway and N. J. A. Sloane, Dec 31 2007

a(A221221(n)) = A133500(A221221(n)) = A222493(n). - Reinhard Zumkeller, May 27 2013

			E.g. 240 -> (0)42 -> 4^2 = 16; 12345 -> 54321 -> 5^4*3^2*1 = 5625.

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

Cf. A131571 (fixed points), A133059 and A133134 (records); A133500 (powertrain).
Cf. A133144 (length of trajectory), A031346 and A003001 (persistence).
Cf. A031298.

a133048 0 = 0
a133048 n = train $ dropWhile (== 0) $ a031298_row n where
   train []       = 1
   train [x]      = x
   train (u:v:ws) = u ^ v * (train ws)
-- Reinhard Zumkeller, May 27 2013

powerback:=proc(n) local a,i,j,t1,t2,t3;
if n = 0 then RETURN(0); fi;
t1:=convert(n, base, 10); t2:=nops(t1);
for i from 1 to t2 do if t1[i] > 0 then break; fi; od:
a:=1; t3:=t2-i+1;
for j from 0 to floor(t3/2)-1 do a := a*t1[i+2*j]^t1[i+2*j+1]; od:
if t3 mod 2 = 1 then a:=a*t1[t2]; fi;
RETURN(a); end;

ptm[n_]:=Module[{idn=IntegerDigits[IntegerReverse[n]]},If[ EvenQ[ Length[idn]],Times@@ (#[[1]]^#[[2]]&/@Partition[idn,2]),(Times@@(#[[1]]^#[[2]]&/@Partition[ Most[ idn],2]))Last[idn]]];Array[ptm,70,0] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 05 2020 *)

A221221 Where powerbacks and powertrains coincide.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 24, 33, 42, 44, 55, 66, 77, 88, 99, 101, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 131, 141, 151, 161, 171, 181, 191, 202, 211, 212, 213, 214, 215, 216, 217, 218, 219, 222, 232, 242, 252, 262, 272, 282, 292

Offset: 1

Reinhard Zumkeller, May 27 2013

Numbers m such that A133048(m) = A133500(m);
A133500(a(n)) = A133048(a(n)) = A222493(n);
if m is a term then also its reversal in decimal representation, palindromes are a subsequence, cf. A004086, A002113.

			Some non-palindromic terms:
a(11) = 10: A133500(10) = 1^0 = 1 = A133048(10) = A133048(1) = 1;
a(14) = 24: A133500(24) = 2^4 = 16 = A133048(24) = 4^2;
a(16) = 42: A133500(42) = 4^2 = 16 = A133048(42) = 2^4;
a(25) = 112: A133500(112) = 1^1 * 2 = 2 = A133048(112) = 2^1 * 1;
a(26) = 113: A133500(113) = 1^1 * 3 = 3 = A133048(113) = 3^1 * 1;
a(44) = 213: A133500(213) = 2^1 * 3 = 6 = A133048(213) = 3^1 * 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

a221221 n = a221221_list !! (n-1)
a221221_list = filter (\x -> a133500 x == a133048 x) [0..]

cp's OEIS Frontend

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^bc^d ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.

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A133048 Powerback(n): reverse the decimal expansion of n, drop any leading zeros, then apply the powertrain map of A133500 to the resulting number.

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A221221 Where powerbacks and powertrains coincide.

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cp's OEIS Frontend

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.

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A133048 Powerback(n): reverse the decimal expansion of n, drop any leading zeros, then apply the powertrain map of A133500 to the resulting number.

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Haskell

Maple

Mathematica

A221221 Where powerbacks and powertrains coincide.

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Haskell

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^bc^d ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.