A135384 -id:A135384 - 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

A309385 Numbers which converge to 24547284284866560000000000 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

6672898, 6689728, 7266898, 7289668, 8966728, 8972668, 23667289, 23668972, 23726689, 23728966, 23896672, 23897266, 26667288, 26668872, 26726688, 26728866, 26886672, 26887266, 29367289, 29368972, 29667287, 29668772, 29723689, 29726687, 29728766, 29728936, 29728993, 29729389, 29876672, 29877266, 29893672, 29897236, 29897293, 29899372, 29937289, 29938972

Offset: 1

Martin Renner, Jul 27 2019

			6672898 -> 6^6*7^2*8^9*8 = 2454728428486656 -> 2^4*5^4*7^2*8^4*2^8*4^8*6^6*5^6 = 24547284284866560000000000.

Martin Renner, Table of n, a(n) for n = 1..702

Cf. A133500, A135384, A135385, A326733, A326734, A326735, A326736, A326737, A326738, A326739, A326740, A326741, A326742.

A326735 Numbers which converge to 2 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

2, 21, 28, 36, 44, 62, 93, 102, 112, 122, 132, 142, 152, 162, 172, 182, 192, 202, 211, 227, 229, 247, 256, 258, 263, 264, 272, 281, 286, 293, 302, 317, 324, 336, 342, 344, 349, 353, 358, 361, 376, 382, 402, 417, 419, 427, 433, 434, 441, 446, 456, 497, 502, 552

Offset: 1

Martin Renner, Jul 22 2019

			28 -> 2^8 = 256 -> 2^5*6 = 192 -> 1^9*2 = 2.

Cf. A133500, A133501, A135384, A135385, A309385, A326733, A326734, A326736, A326737, A326738, A326739, A326740, A326741, A326742.

# change target for A326733-A326742, A135384, A309385
def powertrain(n):
  p, s = 1, str(n)
  if len(s)%2 == 1: s += '1'
  for b, e in zip(s[0::2], s[1::2]): p *= int(b)**int(e)
  return p
def aupto(limit, target=0):
  alst = []
  for n in range(1, limit+1):
    m, ptm = n, powertrain(n)
    while m != ptm: m, ptm = ptm, powertrain(ptm)
    if m == target: alst.append(n)
  return alst
print(aupto(552, target=2)) # Michael S. Branicky, Feb 21 2021

A326733 Numbers which converge to 0 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

0, 26, 35, 37, 38, 39, 43, 46, 47, 48, 49, 54, 55, 56, 57, 58, 59, 64, 65, 66, 67, 68, 69, 72, 73, 74, 76, 77, 78, 79, 82, 84, 85, 86, 87, 88, 89, 94, 95, 96, 97, 98, 99, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 236, 237, 238, 239

Offset: 1

Martin Renner, Jul 22 2019

			26 -> 2^6 = 64 -> 6^4 = 1296 -> 1^2*9^6 = 531441 -> 5^3*1^4*4^1 = 500 -> 5^0*0 = 0.

Cf. A133500, A133501, A135384, A135385, A309385, A326734, A326735, A326736, A326737, A326738, A326739, A326740, A326741, A326742.

A326734 Numbers which converge to 1 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24, 29, 30, 40, 42, 45, 50, 60, 63, 70, 80, 83, 90, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 201, 215, 216, 217, 218, 219, 223, 224, 225, 226, 232, 233, 235, 241, 245, 257, 268, 274, 278, 282, 284, 285

Offset: 1

Martin Renner, Jul 22 2019

			24 -> 2^4 = 16 -> 1^6 = 1.

Cf. A133500, A133501, A135384, A135385, A309385, A326733, A326735, A326736, A326737, A326738, A326739, A326740, A326741, A326742.

A326736 Numbers which converge to 3 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

3, 31, 103, 113, 123, 133, 143, 153, 163, 173, 183, 193, 203, 303, 311, 403, 503, 603, 703, 803, 903, 1031, 1131, 1231, 1331, 1431, 1531, 1631, 1731, 1831, 1931, 2031, 3031, 3100, 3110, 3111, 3112, 3113, 3114, 3115, 3116, 3117, 3118, 3119, 3120, 3130, 3140

Offset: 1

Martin Renner, Jul 22 2019

			31 -> 3^1 = 3.

Cf. A133500, A133501, A135384, A135385, A309385, A326733, A326734, A326735, A326737, A326738, A326739, A326740, A326741, A326742.

A326737 Numbers which converge to 4 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

4, 22, 41, 104, 114, 124, 134, 144, 154, 164, 174, 184, 194, 204, 212, 221, 249, 304, 404, 411, 429, 504, 604, 624, 704, 804, 904, 1022, 1041, 1122, 1141, 1222, 1241, 1322, 1341, 1422, 1441, 1522, 1541, 1622, 1641, 1722, 1741, 1822, 1841, 1922, 1941, 2022

Offset: 1

Martin Renner, Jul 22 2019

			249 -> 2^4*9 = 144 -> 1^4*4 = 4.

Cf. A133500, A133501, A135384, A135385, A309385, A326733, A326734, A326735, A326736, A326738, A326739, A326740, A326741, A326742.

A326738 Numbers which converge to 5 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

5, 51, 53, 105, 115, 125, 135, 145, 155, 165, 175, 185, 195, 205, 305, 335, 345, 405, 455, 505, 511, 525, 527, 531, 537, 605, 692, 705, 745, 805, 875, 905, 925, 1051, 1053, 1151, 1153, 1251, 1253, 1351, 1353, 1451, 1453, 1551, 1553, 1651, 1653, 1751, 1753

Offset: 1

Martin Renner, Jul 22 2019

			53 -> 5^3 = 125 -> 1^2*5 = 5.

Cf. A133500, A133501, A135384, A135385, A309385, A326733, A326734, A326735, A326736, A326737, A326739, A326740, A326741, A326742.

A326739 Numbers which converge to 6 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

6, 61, 106, 116, 126, 136, 146, 156, 166, 176, 186, 196, 206, 213, 306, 312, 406, 506, 606, 611, 706, 724, 746, 806, 906, 1061, 1161, 1261, 1361, 1461, 1561, 1661, 1761, 1861, 1961, 2061, 2131, 2272, 2596, 3061, 3121, 3765, 3863, 4061, 4172, 4572, 5061, 6061

Offset: 1

Martin Renner, Jul 22 2019

			724 -> 7^2*4 = 196 -> 1^9*6 = 6.

Cf. A133500, A133501, A135384, A135385, A309385, A326733, A326734, A326735, A326736, A326737, A326738, A326740, A326741, A326742.

A326740 Numbers which converge to 7 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

7, 71, 75, 107, 117, 127, 137, 147, 157, 167, 177, 187, 197, 207, 307, 407, 507, 523, 543, 607, 707, 711, 723, 747, 751, 807, 907, 1071, 1075, 1171, 1175, 1271, 1275, 1371, 1375, 1471, 1475, 1571, 1575, 1671, 1675, 1771, 1775, 1871, 1875, 1971, 1975, 2071

Offset: 1

Martin Renner, Jul 22 2019

			75 -> 7^5 = 16807 -> 1^6*8^0*7 = 7.

Cf. A133500, A133501, A135384, A135385, A309385, A326733, A326734, A326735, A326736, A326737, A326738, A326739, A326741, A326742.

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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A309385 Numbers which converge to 24547284284866560000000000 under repeated application of the powertrain map of A133500.

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A326735 Numbers which converge to 2 under repeated application of the powertrain map of A133500.

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A326733 Numbers which converge to 0 under repeated application of the powertrain map of A133500.

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A326734 Numbers which converge to 1 under repeated application of the powertrain map of A133500.

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A326736 Numbers which converge to 3 under repeated application of the powertrain map of A133500.

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A326737 Numbers which converge to 4 under repeated application of the powertrain map of A133500.

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A326738 Numbers which converge to 5 under repeated application of the powertrain map of A133500.

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A326739 Numbers which converge to 6 under repeated application of the powertrain map of A133500.

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A326740 Numbers which converge to 7 under repeated application of the powertrain map of A133500.

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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.