A064869 -id:A064869 - cp's OEIS frontend

A067159 Number of regions in regular n-gon which are 12-gons.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 51, 0, 53, 0, 0, 0, 0, 0, 59, 0, 61, 0, 63, 0, 65, 0, 0, 0, 0, 0, 0, 0, 73, 0, 225, 76, 0, 0, 0, 0, 0, 0, 0, 0, 170, 0, 0, 0, 0, 0, 273, 0, 93, 0, 95, 0, 97, 0, 0, 0, 101, 102, 412

Offset: 12

Sascha Kurz, Jan 06 2002

nonn

			a(40)=40 because drawing the regular 40-gon with all its diagonals yields 40 12-gons.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 12..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A064869, A067151, A067152, A067153, A067154, A067155, A067156, A067157, A067158.

a(104) and beyond by Scott R. Shannon, Dec 04 2021

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

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 *)

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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A067159 Number of regions in regular n-gon which are 12-gons.

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

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Maple

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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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