A064872 -id:A064872 - 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)

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

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.

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

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

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

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

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

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

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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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