A064868 -id:A064868 - cp's OEIS frontend

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

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.

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)

A064870 The minimal number which has multiplicative persistence 6 in base n.

Original entry on oeis.org

11262, 57596799, 30536, 6788, 4684, 1571, 439, 667, 1964, 683, 218, 857, 264, 278, 353, 393, 227, 382, 344, 311, 319, 307, 283, 417, 422, 381, 485, 436, 349, 431, 436, 449, 421, 469, 327, 575, 598, 483, 539, 413, 511, 517, 534, 641, 611, 609, 476, 479

Offset: 7

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(5)=1811981201171874, a(6) seems not to exist.

			a(13) = 439 because 439 = [2'7'10]->[10'10]->[7'9]->[4'11]->[3'5]->[1'2]->[2] needs 6 steps and no fewer n.

Michael De Vlieger, Table of n, a(n) for n = 7..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, order 721.

Cf. A003001, A031346, A064867, A064868, A064869, A064871, A064872.

a(n) = 7*n-[n/720] for n > 719.

A064872 The minimal number which has multiplicative persistence 8 in base n.

Original entry on oeis.org

7577, 130883, 596667, 3644381, 2820, 61773, 2752, 5136, 7452, 38631, 2780, 8015, 2996, 542, 8611, 4591, 575, 10586, 2532, 2681, 2764, 1016, 4547, 10151, 1065, 983, 813, 5431, 900, 1255, 983, 5179, 5117, 1190, 982, 1129, 1501, 1491, 1471, 1084

Offset: 13

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(7)=1086400325525346, a(10)=2677889, a(11)=757074, a(8) and a(9) seem not to exist.

			a(13) = 7577 because 7577 is the fewest number with persistence 8 in base 13.

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, order 40321.

Cf. A003001, A031346, A064867, A064868, A064869, A064870, A064871.

a(n) = 9*n-[n/40320] for n > 40319.

A064871 The minimal number which has multiplicative persistence 7 in base n.

Original entry on oeis.org

1409794, 68889, 38200, 17902874277, 1494, 2532, 19526, 15838, 1101, 15820, 943, 2674, 2118, 3275, 412, 3310, 1593, 440, 478, 2036, 456, 713, 738, 633, 658, 705, 907, 643, 803, 641, 653, 797, 484, 991, 814, 877, 1079, 767, 840, 575, 930, 843, 710, 880

Offset: 9

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(7) = 686285, a(8) seems not to exist.

			a(9) = 1409794 because the persistence of 1409794 is 7.

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, order 5041.

Cf. A003001, A031346, A064867, A064868, A064869, A064870, A064872.

a(n) = 8*n-[n/5040] for n > 5039.

Corrected by R. J. Mathar, Nov 02 2007

A245760 Maximal multiplicative persistence of n in any base.

Original entry on oeis.org

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

Offset: 1

Sergio Pimentel, Jul 31 2014

nonn

It has been conjectured that there is a maximum multiplicative persistence in a given base, but it is not known if this sequence is bounded.

In fact, Theorem 1 in Lamont-Smith paper implies that this sequence is unbounded. - Brendan Gimby, Jul 12 2025

			a(23)=3 since the persistence of 23 in base 6 is 3 (23 in base 6 is 35 / 3x5=15 / 15 in base 6 is 23 / 2x3=6 / 6 in base 6 is 10 / 1x0=0 which is a single digit). In any other base the persistence of 23 is 3 or less, therefore a(23)=3.
a(12)=1 since 12 does not have a multiplicative persistence greater than 1 in any base.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Tim Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.

Cf. A003001, A031346, A064867, A064868, A046510.

persistence:= proc(n,b) local i,m;
  m:= n;
  for i from 1 do
       m:= convert(convert(m,base,b),`*`);
     if m < b then return i fi
  od:
end proc:
A:= n -> max(seq(persistence(n,b),b=2..n-1)):
0, 1, seq(A(n),n=3..100); # Robert Israel, Jul 31 2014

persistence[n_, b_] := Module[{i, m}, m = n; For[i = 1, True, i++, m = Times @@ IntegerDigits[m, b]; If[m < b, Return [i]]]];
A[n_] := Max[Table[persistence[n, b], {b, 2, n-1}]];
Join[{0, 1}, Table[A[n], {n, 3, 100}]] (* Jean-François Alcover, Apr 30 2019, after Robert Israel *)

A125582 Smallest positive integer with multiplicative persistence n in base 12.

Original entry on oeis.org

1, 12, 30, 46, 83, 1099, 1571, 17902874277

Offset: 0

Walter Kehowski, Jan 04 2007

The sequence in base 12 is 1, 10, 26, 3X, 6E, 777, XXE, 3577777799, where X is 10 and E is 11. I have searched numbers up to 24 digits in base 12 excluding any numbers that might contain the digit 1 or any combination of digits that might multiply to 0 mod 12. The numbers also had digits in nondecreasing order, so that XXE would be tested but, for example, EXX would not.

			a(0)=1 since 1 is the smallest positive integer for which no multiplication takes place. [Edited by _A.H.M. Smeets_, Sep 16 2018]
a(6)=1571 since 1571, 1100, 392, 128, 80, 48, 0 is the chain with six multiplications. In base 12, XXE, 778, 288, X8, 68, 40, 0.

Cf. A003001, A031346, A046510, A064867, A064868, A064869, A064870, A064871, A064872.

Maple
```
Maple program available upon request.
```

With[{s = Array[-1 + Length@ FixedPointList[Times @@ IntegerDigits[#, 12] &, #] &, 1600]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Sep 18 2018 *)

A132161 Smallest number which has multiplicative persistence n in base 16.

Original entry on oeis.org

1, 16, 40, 62, 95, 187, 683, 15838, 3644381

Offset: 0

R. J. Mathar, Nov 02 2007

See A003001 for base 10 and A125582 for base 12. a(3), a(4),...,a(8) are A064867(16), A064868(16),...,A064872(16).

a(8) drawn from A064872 by Michel Marcus, Jul 23 2013

A330152 Absolute multiplicative persistence: a(n) is the least number with multiplicative persistence n for some base b > 1.

Original entry on oeis.org

0, 2, 8, 23, 52, 127, 218, 412, 542, 692, 1471, 2064, 2327, 4739, 13025, 16213, 20388, 45407, 82605, 123706, 207778, 323382, 605338, 905670, 1033731, 2041995, 3325970, 4282238, 7638962, 9840138, 10364329, 26110715, 40706834, 57222153, 82242809, 97900397

Offset: 0

Tim Lamont-Smith, Nov 29 2019

			2 when represented in base 2 goes 10 -> 0 and has an absolute persistence of 1, so a(1) = 2.
8 when represented in base 3 goes 22 -> 11 -> 1 and has an absolute persistence of 2, so a(2) = 8.
23 when represented in base 6 goes 35 -> 23 -> 10 -> 1 and has absolute persistence of 3, so a(3) = 23 (Cf. A064867).
52 when represented in base 9 goes 57 -> 38 -> 26 -> 13 -> 3 and has absolute persistence of 4, so a(4) = 52 (Cf. A064868).

Edson de Faria and Charles Tresser, On Sloane's persistence problem, arXiv preprint arXiv:1307.1188 [math.DS], 2013.
Edson de Faria and Charles Tresser, On Sloane's persistence problem, Experimental Math., 23 (No. 4, 2014), 363-382.
Brendan Gimby, Tools for finding numbers with large persistence.
Brady Haran and Matt Parker, What's special about 277777788888899?, Numberphile video, 2019.
Tim Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
Stephanie Perez and Robert Styer, Persistence: A Digit Problem, Involve, Vol. 8 (2015), No. 3, 439-446.
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, A064867, A064868, A064869, A064870, A064871, A064872.

from math import prod
from sympy.ntheory.digits import digits
def mp(n, b): # multiplicative persistence of n in base b
    c = 0
    while n >= b:
        n, c = prod(digits(n, b)[1:]), c+1
    return c
def a(n):
    k = 0
    while True:
        if any(mp(k, b)==n for b in range(2, max(3, k))): return k
        k += 1
print([a(n) for n in range(11)]) # Michael S. Branicky, Sep 17 2021

a(19)-a(27) from Giovanni Resta, Jan 20 2020
a(28)-a(30) from Michael S. Branicky, Sep 17 2021
a(31)-a(35) from Brendan Gimby, Jul 08 2025

A245761 Numbers with a maximal multiplicative persistence of 1 in any base.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 12

Offset: 1

Sergio Pimentel, Jul 31 2014

			12 is in the sequence since the persistence of 12 is at most 1 in any base. I.e. it takes at most one step to go from 12 to a single digit in any base, e.g., in base 2 we have 1100 -> 0. In base 5 we have 22 -> 4. In bases 12 and above the initial number is already a single digit.
This sequence is complete - there are no other terms. - _Alois P. Heinz_, Jul 31 2014

Cf. A003001, A031346, A064867, A064868, A046510, A245760.

cp's OEIS Frontend

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

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

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A064870 The minimal number which has multiplicative persistence 6 in base n.

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A064872 The minimal number which has multiplicative persistence 8 in base n.

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A064871 The minimal number which has multiplicative persistence 7 in base n.

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A245760 Maximal multiplicative persistence of n in any base.

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Maple

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A125582 Smallest positive integer with multiplicative persistence n in base 12.

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Maple

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A132161 Smallest number which has multiplicative persistence n in base 16.

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A330152 Absolute multiplicative persistence: a(n) is the least number with multiplicative persistence n for some base b > 1.

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