A003001 -id:A003001 - cp's OEIS frontend

A031347 Multiplicative digital root of n (keep multiplying digits of n until reaching a single digit).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0, 4, 8, 2, 6, 0, 8, 6, 6, 8, 0, 5, 0, 5, 0, 0, 0, 5, 0, 0, 0, 6, 2, 8, 8, 0, 8, 8, 6, 0, 0, 7, 4, 2, 6, 5, 8, 8, 0, 8, 0, 8, 6, 8, 6, 0, 6

Offset: 0

Eric W. Weisstein

a(n) = 0 for almost all n. - Charles R Greathouse IV, Oct 02 2013

More precisely, a(n) = 0 asymptotically almost surely, namely, among others, for all numbers n which have a digit '0', and as n has more and more digits, it becomes increasingly less probable that no digit is equal to zero. (The set A011540 has density 1.) Thus the density of numbers for which a(n) > 0 is zero, although this happens for infinitely many numbers, for example all repunits n = (10^k - 1)/9 = A002275(k). - M. F. Hasler, Oct 11 2015

T. D. Noe, Table of n, a(n) for n = 0..10000
Shyam Sunder Gupta, Digital Root Wonders, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 1, 1-28.
Eric Weisstein's World of Mathematics, Multiplicative Digital Root.
Index entries for Colombian or self numbers and related sequences

Cf. A007954, A007953, A003001, A010888 (additive digital root of n), A031286 (additive persistence of n), A031346 (multiplicative persistence of n).

Numbers having multiplicative digital roots 0-9: A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056.

a031347 = until (< 10) a007954
-- Reinhard Zumkeller, Oct 17 2011, Sep 22 2011

A007954 := proc(n) return mul(d, d=convert(n,base,10)): end: A031347 := proc(n) local m: m:=n: while(length(m)>1)do m:=A007954(m): od: return m: end: seq(A031347(n),n=0..100); # Nathaniel Johnston, May 04 2011

mdr[n_] := NestWhile[Times @@ IntegerDigits@# &, n, UnsameQ, All]; Table[ mdr[n], {n, 0, 104}] (* Robert G. Wilson v, Aug 04 2006 *)
Table[NestWhile[Times@@IntegerDigits[#] &, n, # > 9 &], {n, 0, 90}] (* Harvey P. Dale, Mar 10 2019 *)

A031347(n)=local(resul); if(n<10, return(n) ); resul = n % 10; n = (n - n%10)/10; while( n > 0, resul *= n %10; n = (n - n%10)/10; ); return(A031347(resul))
for(n=1,80, print1(A031347(n),",")) \\ R. J. Mathar, May 23 2006

A031347(n)={while(n>9,n=prod(i=1,#n=digits(n),n[i]));n} \\ M. F. Hasler, Dec 07 2014

from operator import mul
from functools import reduce
def A031347(n):
    while n > 9:
       n = reduce(mul, (int(d) for d in str(n)))
    return n
# Chai Wah Wu, Aug 23 2014

from math import prod
def A031347(n):
    while n > 9: n = prod(map(int, str(n)))
    return n
print([A031347(n) for n in range(100)]) # Michael S. Branicky, Apr 17 2024

def iterDigitProd(n: Int): Int = n.toString.length match {
  case 1 => n
  case  => iterDigitProd(n.toString.toCharArray.map( - 48).scanRight(1)( * ).head)
}
(0 to 99).map(iterDigitProd) // Alonso del Arte, Apr 11 2020

a(n) = d in {1, ..., 9} if (but not only if) n = (10^k - 1)/9 + (d - 1)*10^m = A002275(k) + (d - 1)*A011557(m) for some k > m >= 0. - M. F. Hasler, Oct 11 2015

A031346 Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 1, 1, 2, 2, 2, 3, 2, 3, 2, 3, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 1, 1, 2, 2, 3, 3, 2, 4, 3, 3, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 2, 3, 3, 3, 3, 3, 3, 2

Offset: 0

Eric W. Weisstein

			For n = 999: A007954(999) = 729, A007954(729) = 126, A007954(126) = 12 and A007954(12) = 2. The fourth iteration of "multiply digits" yields a single-digit number, so a(999) = 4. - _Felix Fröhlich_, Jul 17 2016

M. Gardner, Fractal Music, Hypercards and More Mathematical Recreations from Scientific American, Persistence of Numbers, pp. 120-1; 186-7, W. H. Freeman NY 1992.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 35.

T. D. Noe, Table of n, a(n) for n = 0..10000
Gabriel Bonuccelli, Lucas Colucci, and Edson de Faria, On the Erdős-Sloane and Shifted Sloane Persistence, arXiv:2009.01114 [math.NT], 2020.
Eric Brier, Christophe Clavier, Linda Gutsche and David Naccache, The Multiplicative Persistence Conjecture Is True for Odd Targets, arXiv:2110.04263 [math.NT], 2021.
M. R. Diamond, Multiplicative persistence base 10: some new null results.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence.

Cf. A007954 (product of decimal digits of n).
Cf. A010888 (additive digital root of n).
Cf. A031286 (additive persistence of n).
Cf. A031347 (multiplicative digital root of n).
Cf. A263131 (ordinal transform).
Cf. A003001.

f:=func; a:=[]; for n in [0..100] do s:=0; k:=n; while k ge 10 do s:=s+1; k:=f(k); end while; Append(~a,s); end for; a; // Marius A. Burtea, Jan 12 2020

A007954 := proc(n) return mul(d, d=convert(n, base, 10)): end: A031346 := proc(n) local k,m: k:=0:m:=n: while(length(m)>1)do m:=A007954(m):k:=k+1: od: return k: end: seq(A031346(n),n=0..100); # Nathaniel Johnston, May 04 2011

Table[Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>=10&]],{n,0,100}]-1 (* Harvey P. Dale, Aug 27 2016 *)

a007954(n) = my(d=digits(n)); prod(i=1, #d, d[i])
a(n) = my(k=n, i=0); while(#Str(k) > 1, k=a007954(k); i++); i \\ Felix Fröhlich, Jul 17 2016

from operator import mul
from functools import reduce
def A031346(n):
    mp = 0
    while n > 9:
        n = reduce(mul, (int(d) for d in str(n)))
        mp += 1
    return mp
# Chai Wah Wu, Aug 23 2014

Probably bounded, see A003001. - Charles R Greathouse IV, Nov 15 2022

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.

Original entry on oeis.org

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

A064869 The minimal number which has multiplicative persistence 5 in base n.

Original entry on oeis.org

244140624, 3629, 1601, 1535, 394, 679, 317, 1099, 127, 135, 582, 187, 168, 157, 201, 159, 230, 215, 180, 185, 246, 181, 188, 195, 198, 323, 239, 255, 259, 267, 239, 287, 295, 293, 310, 313, 280, 377, 375, 395, 347, 360, 321, 370, 439, 431, 458, 355, 362

Offset: 5

Sascha Kurz, Oct 09 2001

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit. a(3) and a(4) seem not to exist.

			a(9)=394 because 394=[477]->[237]->[46]->[26]->[13]->[3] and no smaller n has persistence 5 in base 9.

Michael De Vlieger, Table of n, a(n) for n = 5..10000
M. R. Diamond and D. D. Reidpath, A counterexample to a conjecture of Sloane and Erdos, J. Recreational Math., 1998 29(2), 89-92.
Sascha Kurz, Persistence in different bases
T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
Carlos Rivera, Puzzle 22. Primes and Persistence, The Prime Puzzles and Problems Connection.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for linear recurrences with constant coefficients, order 121.

Cf. A003001, A031346, A064867, A064868, A064870, A064871, A064872.

a(n) = 6*n-floor(n/120) for n > 119.

A006050 Smallest number of additive persistence n.

Original entry on oeis.org

0, 10, 19, 199, 19999999999999999999999

Offset: 0

N. J. A. Sloane

The next term a(5) is 1 followed by 2222222222222222222222 9's.

Meimaris Antonios, On the additive persistence of a number in base p, Preprint, 2015.
H. J. Hindin, The additive persistence of a number, J. Rec. Math., 7 (No. 2, 1974), 134-135.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Shyam Sunder Gupta, Digital Root Wonders, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 1, 1-28.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Additive Persistence.

Cf. A003001, A031286, A045646.

lst = {0, 10}; Do[AppendTo[lst, 2*10^((lst[[-1]] - 1)/9) - 1], {3}]; lst (* Arkadiusz Wesolowski, Oct 17 2012 *)
Join[{0},NestList[2*10^((#-1)/9)-1&,10,3]] (* Harvey P. Dale, May 08 2020 *)

For n>1 a(n) = 2*10^((a(n-1)-1)/9)-1.

A014120 Smallest number of persistence n over product-of-nonzero-digits function.

Original entry on oeis.org

0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 267777777889999, 77777777788888888888899999, 37777777777777777777777777778888889999999999999999999

Offset: 0

David W. Wilson

Comments from Marc Lapierre, Sep 09 2020, updated Feb 19 2025: (Start)
Let d(b) denote a string of b copies of the digit d. For example, 3(5) would represent 33333.
Here are recent discoveries and the initials of the discoverers:
2(1)6(1)7(1)8(99)9(10) by WW
6(1)7(157)8(46)9(25) by WW
3(1)7(54)8(82)9(353) by WW
3(1)7(27)8(622)9(399) by WW
3(1)7(140)8(258)9(1946) by WW
2(1)7(122)8(498)9(4297) by ML
2(1)7(723)8(211)9(9825) by ML
Conjectured terms follow:
2(1)6(1)7(822)8(29)9(22601) by CC
2(1)7(325)8(1678)9(49461) by CC
2(1)7(1199)8(662)9(110077) by CC
7(1335)8(1609)9(241857) by CC
4(1)7(155)8(1135)9(540199) by CC
6(1)7(124)8(762)9(1192494) by CC
4(1)7(94)8(583)9(2616367) by CC
3(1)5(6)9(5788986) by CC
7(20)8(24)9(12878515) by CC
8(67)9(29136478) by CC
7(9)8(17)9(64768493) by CC
6(1)7(8)8(5)9(144417969) by CC
6(1)7(2)8(8)9(324379548) by CC
2(1)6(1)7(1)8(31)9(728759871) by CC
2(1)6(1)8(555702)9(1553958443) by CC
3(1)8(10311016)9(3652133953) by CC
2(1)6(1)8(1482224)9(9136624827) by CC
3(1)8(1469099)9(19262606200) by CC
8(30408394)9(42905304588) by CC
WW=Wilfred Whiteside & Phil Carmody (see link)
ML=Marc Lapierre
CC=Christophe Clavier
It seems a safe conjecture that this sequence is infinite.
(End)

Marc Lapierre, Table of n, a(n) for n = 0..16
Marc Lapierre, Multiplicative persistence computation
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Wilfred Whiteside & Phil Carmody, Puzzle 341. Multiplicative persistence, Erdos style, Problems & Puzzles: Puzzles.

Cf. A003001.

Edited by N. J. A. Sloane, Sep 11 2020 following suggestions from Marc Lapierre, Sep 09 2020

A064867 The minimal number which has multiplicative persistence 3 in base n.

Original entry on oeis.org

26, 63, 68, 23, 27, 31, 35, 39, 43, 46, 50, 54, 58, 62, 66, 69, 73, 77, 81, 85, 89, 92, 96, 100, 104, 108, 112, 115, 119, 123, 127, 131, 135, 138, 142, 146, 150, 154, 158, 161, 165, 169, 173, 177, 181, 184, 188, 192

Offset: 3

Sascha Kurz, Oct 08 2001

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit.

			a(3) = 26 because 26 = [222]->[22]->[11]->[1] and no fewer n has persistence 3 in base 3.

Michael De Vlieger, Table of n, a(n) for n = 3..10000
M. R. Diamond and D. D. Reidpath, A counterexample to a conjecture of Sloane and Erdos, J. Recreational Math., 1998 29(2), 89-92.
Sascha Kurz, Persistence in different bases
T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
C. Rivera, Minimal prime with persistence p
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A003001, A031346, A064868, A064869, A064870, A064871, A064872.

With[{m = 3}, Table[Block[{k = 1}, While[Length@ FixedPointList[Times @@ IntegerDigits[#, n] &, k, 100] != m + 2, k++]; k], {n, 3, 5}]]~Join~Array[4 # - Floor[#/6] &, 45, 6] (* Michael De Vlieger, Aug 30 2021 *)

a(n) = 4*n-floor(n/6) for n > 5.
From Chai Wah Wu, Mar 07 2025: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n > 12.
G.f.: x^3*(48*x^9 - x^8 - 33*x^7 - 22*x^6 + 4*x^5 + 4*x^4 - 45*x^3 + 5*x^2 + 37*x + 26)/(x^7 - x^6 - x + 1). (End)

A064868 The minimal number which has multiplicative persistence 4 in base n.

Original entry on oeis.org

2344, 172, 131, 174, 52, 77, 75, 83, 75, 81, 89, 95, 101, 104, 110, 133, 143, 127, 133, 119, 124, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 238, 243, 248, 253, 258, 263, 268, 273, 278, 283

Offset: 5

Sascha Kurz, Oct 09 2001

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit. a(3) and a(4) do not seem to exist.

			a(6) = 172 because 172 = [444]->[144]->[24]->[12]->[2] and no lesser n has persistence 4 in base 6.

Michael De Vlieger, Table of n, a(n) for n = 5..10000
M. R. Diamond and D. D. Reidpath, A counterexample to a conjecture of Sloane and Erdos, J. Recreational Math., 1998 29(2), 89-92.
Sascha Kurz, Persistence in different bases
T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
C. Rivera, Minimal prime with persistence p
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A003001, A031346, A064867, A064869, A064870, A064871, A064872.

With[{m = 4, r = 24}, Table[Block[{k = 1}, While[Length@ FixedPointList[Times @@ IntegerDigits[#, n] &, k] != m + 2, k++]; k], {n, m + 1, r}]~Join~Array[(m + 1) # - Floor[#/r] &, 34, r + 1]] (* Michael De Vlieger, Aug 30 2021 *)

pers(nn, b) = {ok = 0; p = 0; until (ok, d = digits(nn, b); if (#d == 1, ok = 1, p++); nn = prod(k=1, #d, d[k]); if (nn == 0, ok = 1);); return (p);}
a(n) = {i=0; while (pers(i, n) != 4, i++); return (i);} \\ Michel Marcus, Jun 30 2013

a(n) = 5*n-floor(n/24) for n > 23.
From Chai Wah Wu, Mar 07 2025: (Start)
a(n) = a(n-1) + a(n-24) - a(n-25) for n > 48.
G.f.: x^5*(18*x^43 - x^42 + 21*x^41 - 5*x^40 - 18*x^39 - x^38 + 2*x^37 - x^36 - x^35 - 3*x^34 - x^33 + 13*x^32 - 3*x^31 + 7*x^30 - 20*x^29 + 127*x^28 - 38*x^27 + 46*x^26 + 2177*x^25 - 2339*x^24 + 5*x^23 + 5*x^22 + 5*x^21 + 5*x^20 - 14*x^19 + 6*x^18 - 16*x^17 + 10*x^16 + 23*x^15 + 6*x^14 + 3*x^13 + 6*x^12 + 6*x^11 + 8*x^10 + 6*x^9 - 8*x^8 + 8*x^7 - 2*x^6 + 25*x^5 - 122*x^4 + 43*x^3 - 41*x^2 - 2172*x + 2344)/(x^25 - x^24 - x + 1). (End)

Example modified by Harvey P. Dale, Oct 19 2022

A046500 Smallest prime with multiplicative persistence n.

Original entry on oeis.org

2, 11, 29, 47, 277, 769, 8867, 186889, 2678789, 26899889, 3778888999, 277777788888989

Offset: 0

Patrick De Geest, Sep 15 1998

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit.

			47 -> 28 -> 16 -> 6 has persistence 3.

Shyam Sunder Gupta, Digital Root Wonders, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 1, 1-28.
Carlos Rivera, Puzzle 22. Primes & Persistence, The Prime Puzzles & Problems Connection.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence.

Cf. A003001, A014120.

a[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]-1; t={}; i=1; Do[While[a[p=Prime[i]]!=n,i++]; AppendTo[t,p],{n,0,9}]; t (* Jayanta Basu, Jun 02 2013 *)

a(10)-a(11) found by Jud McCranie

A046510 Numbers with multiplicative persistence value 1.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 30, 31, 32, 33, 40, 41, 42, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 130, 131, 132, 133

Offset: 1

Patrick De Geest, Sep 15 1998

Numbers 0 to 9 have a multiplication persistence of 0, not 1. - Daniel Mondot, Mar 12 2022

			24 -> 2 * 4 = [ 8 ] -> one digit in one step.

Daniel Mondot, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046501.

Numbers with multiplicative persistence m: this sequence (m=1), A046511 (m=2), A046512 (m=3), A046513 (m=4), A046514 (m=5), A046515 (m=6), A046516 (m=7), A046517 (m=8), A046518 (m=9), A352531 (m=10), A352532 (m=11).

Select[Range[10, 121], IntegerLength[Times @@ IntegerDigits[#]] <= 1 &] (* Jayanta Basu, Jun 26 2013 *)

isok(n) = my(d=digits(n)); (#d > 1) && (#digits(prod(k=1, #d, d[k])) <= 1); \\ Michel Marcus, Apr 12 2018 and Mar 13 2022

from math import prod
def ok(n): return n > 9 and prod(map(int, str(n))) < 10
print([k for k in range(134) if ok(k)]) # Michael S. Branicky, Mar 13 2022

Incorrect terms 0 to 9 removed by Daniel Mondot, Mar 12 2022

cp's OEIS Frontend

A031347 Multiplicative digital root of n (keep multiplying digits of n until reaching a single digit).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Python

Scala

Formula

A031346 Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

A064869 The minimal number which has multiplicative persistence 5 in base n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A006050 Smallest number of additive persistence n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A014120 Smallest number of persistence n over product-of-nonzero-digits function.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A064867 The minimal number which has multiplicative persistence 3 in base n.

Views

Author

Keywords

Comments

Examples

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.